Equilibrium states on operator algebras associated to self-similar ...

EQUILIBRIUM STATES ON OPERATOR ALGEBRAS ASSOCIATED TO SELF-SIMILAR ACTIONS OF GROUPOIDS ON GRAPHS

arXiv:1610.00343v1 [math.OA] 2 Oct 2016

MARCELO LACA, IAIN RAEBURN, JACQUI RAMAGGE, AND MICHAEL F. WHITTAKER Abstract. We consider self-similar actions of groupoids on the path spaces of finite directed graphs, and construct examples of such self-similar actions using a suitable notion of graph automaton. Self-similar groupoid actions have a Cuntz-Pimsner algebra and a Toeplitz algebra, both of which carry natural dynamics lifted from the gauge actions. We study the equilibrium states (the KMS states) on the resulting dynamical systems. Above a critical inverse temperature, the KMS states on the Toeplitz algebra are parametrised by the traces on the full C ∗ -algebra of the groupoid, and we describe a program for finding such traces. The critical inverse temperature is the logarithm of the spectral radius of the incidence matrix of the graph, and at the critical temperature the KMS states on the Toeplitz algebra factor through states of the Cuntz-Pimsner algebra. Under a verifiable hypothesis on the self-similar action, there is a unique KMS state on the Cuntz-Pimsner algebra. We discuss an explicit method of computing the values of this KMS state, and illustrate with examples.

1. Introduction A self-similar group (G, X) consists of a finite set X and a faithful action of a group G on the set X ∗ of finite words in X, such that: for each g ∈ G and x ∈ X, there exists h ∈ G satisfying g · (xw) = (g · x)(h · w) for all w ∈ X ∗ .

Self-similar groups are often defined recursively using data presented in an automaton (see, for example, [20, Chapter 1] or §2 below). To each self-similar group (G, X), Nekrashevych associated a C ∗ -algebra O(G, X), which is by definition the Cuntz-Pimsner algebra of a Hilbert bimodule over the reduced group algebra Cr∗ (G) [19, 21]. We recently studied the Toeplitz algebra T (G, X) of this Hilbert bimodule [17]. Both T (G, X) and O(G, X) carry natural gauge actions of the unit circle, and composing with the exponential map gives actions α of the real line. In [17], we classified the equilibrium states (the KMS states) of the dynamical systems (T (G, X), R, α) and (O(G, X), R, α). We found a simplex of KMS states on T (G, X) at all inverse temperatures larger than a critical value ln |X|, and showed, under a mild hypothesis on (G, X), that there is a single KMS state on O(G, X) whose inverse temperature is ln |X|. Here we consider a new kind of self-similarity involving an action of a groupoid G on the path space E ∗ of a finite directed graph E, which we view as a forest of trees {vE ∗ : v ∈ E 0 }. An equation of the form g · (eµ) = (g · e)(h · µ)

This research was supported by the Natural Sciences and Engineering Research Council of Canada, the Marsden Fund of the Royal Society of New Zealand, and the Australian Research Council. 1

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MARCELO LACA, IAIN RAEBURN, JACQUI RAMAGGE, AND MICHAEL F. WHITTAKER

for paths eµ defines an isomorphism h of the subtree s(e)E ∗ onto s(g · e)E ∗ . We call h a partial isomorphism of E ∗ . The partial isomorphisms of E ∗ form a groupoid PIso(E ∗ ) with unit space E 0 . A self-similar action of a groupoid G with unit space E 0 is then a groupoid homomorphism of G into PIso(E ∗ ). Our results are motivated by a construction of Exel and Pardo [8], who studied a family of self-similar actions of groups on path spaces. Their main motivation was to provide a unified theory that accommodates both Nekrashevych’s Cuntz-Pimsner algebras and a family of “Katsura algebras” [13] that includes all Kirchberg algebras. We seek a common setting for the analyses of KMS states on self-similar groups in [17] and on the ToeplitzCuntz-Krieger algebras of graphs [7, 12, 10, 11]. To each of our self-similar actions (G, E) we associate a Toeplitz algebra T (G, E) and a Cuntz-Pimsner algebra O(G, E), and study the dynamics arising from the gauge actions of the circle. Above a critical inverse temperature, the KMS states on T (G, E) are parametrised by traces on the coefficient algebra, which is the (full) C ∗ -algebra of the groupoid G. As similar analyses for graph algebras have consistently shown [5, 12, 10], the critical inverse temperature is ln ρ(B), where B is the vertex matrix of the underlying graph E and ρ(B) is the spectral radius of B. At the critical inverse temperature, and under mild hypotheses, the KMSln ρ(B) state is unique and factors through the CuntzPimsner algebra. Thus our results give those of [17] a distinct graph-theoretic slant. On the other hand, we also show how to compute the values of the KMSln ρ(B) state by counting paths in Moore diagrams, as we did for self-similar groups in [17, §8]. Outline. After quickly reviewing the process by which automata give rise to self-similar group actions, we describe our self-similar actions in §3. We follow closely the analogy between the space X ∗ of words, which is often viewed as a rooted tree, and the path space E ∗ , which is a forest of rooted trees with the vertices v ∈ E 0 as roots. Our groupoid acts by partial isomorphisms of this forest, which are isomorphisms of one rooted tree onto another. These partial isomorphisms form a groupoid PIso(E ∗ ) with unit space E 0 (Proposition 3.2). An action of a groupoid with unit space E 0 is then simply a homomorphism of G into PIso(E ∗ ), and our definition of self-similarity is formally just the usual one. After a brief discussion of the basic properties, we introduce a notion of E-automaton, and show how it gives rise a self-similar groupoid action on E ∗ . Our treatment and notation are based on the established formalism for self-similar groups, and hence look quite different from that of Exel and Pardo. So we show in an appendix that their data for a group K gives a self-similar action of the groupoid K × E 0 . But not all of our self-similar groupoid actions arise this way (see Remark 3.5). In §4, we construct the Toeplitz algebra T (G, E) of a self-similar groupoid action (G, E). Our constructions of KMS states follow the strategy developed in [15, 16, 17], and are intrinsically representation-theoretic in nature. To construct representations of our algebras, we use presentations of our algebras. Thus we take as our coefficient algebra the full groupoid algebra C ∗ (G), just as we used the full group algebra in [17] because it is universal for unitary representations of the group. We discuss the universal property of C ∗ (G) in Proposition 4.1. The presentation of the Toeplitz algebra is given in Proposition 4.4: it is generated by a Toeplitz-Cuntz-Krieger family for the graph E and a representation of the groupoid. In Proposition 4.5 we identify a convenient spanning family consisting of elements which are analytic for the natural dynamics.

SELF-SIMILAR ACTIONS OF GROUPOIDS ON GRAPHS

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At this point we are ready to begin our analysis of the KMS states. In Proposition 5.1, we describe how to recognise KMS states in terms of their values on the spanning elements. In Theorem 6.1, we describe the KMSβ states on the Toeplitz algebra for β above the critical inverse temperature ln ρ(B), finding that they are parametrised by the traces on the coefficient algebra C ∗ (G). Thus in §7 we consider the problem of constructing traces on the groupoid algebra C ∗ (G). Since our groupoids have finite unit space E 0 , the orbit space for the canonical action of G on E 0 is finite, and C ∗ (G) is the direct sum of C ∗ algebras of transitive groupoids (see Lemma 7.5). For a transitive groupoid we can realise C ∗ (G) as matrices over the C ∗ -algebra C ∗ (Gx ) of an isotropy group, and then traces on C ∗ (Gx ) give traces on C ∗ (G) (see Corollary 7.6). So there are always at least two traces on each C ∗ (G). We check that in our special cases (that is, for traditional self-similar groups and for graph algebras), the resulting KMS states are indeed the ones previously found in [17] and [10]. In §8, we prove that T (G, E) always has a KMSln ρ(B) state, and that this state factors through O(G, E). Under an easily verified hypothesis on (G, E), this is the only KMSln ρ(B) state on T (G, E) (Theorem 8.3). The proof of uniqueness in particular is analytically quite delicate. We close with a section on examples. Since the formulas for the KMS states in Theorem 8.3 are rather intricate, it is comforting that in concrete examples we can compute particular values of these states, getting some strange and rather wild numbers (see Proposition 9.5). 2. Preamble: Self-similar group actions and automata We begin by reviewing the process by which automata are used to construct self-similar group actions. This process is standard (see [20, Chapter 1], for example), but having the details clear will help us later. Suppose that XFis a finite set, and write X k for the set of k-tuples in X, with X 0 = {∗}, and define X ∗ := k≥0 X k . Consider the (undirected) rooted tree T = TX with vertex set T 0 = X ∗ and edge set T 1 = {{µ, µx} : µ ∈ X ∗ and x ∈ X}. Note that X k is here identified with the vertices in T at depth k from the root ∗. From a traditional graph-theoretic perspective, an automorphism α of T consists of a family of bijections αk : X k → X k for k ≥ 0 such that for all µ, ν ∈ X ∗ (2.1)

{αk (µ), αk+1(ν)} ∈ T 1 ⇔ {µ, ν} ∈ T 1 .

It is sometimes convenient to use an alternative, but equivalent, definition of an automorphism of T that is better suited to properties of self-similar actions. Notice that if β = {βk } is an automorphism, then each {βk (µ), βk+1(µx)} is an edge in T , and hence βk+1 (µx) ∈ βk (µ)X. So an automorphism satisfies (2.2) below, and the property (2.2) is ostensibly weaker. Lemma 2.1. Suppose X is a finite set and T is the rooted tree with vertex set T 0 = X ∗ and edge set T 1 = {{µ, µx} : µ ∈ X ∗ and x ∈ X}. Suppose α : T 0 → T 0 is a bijection satisfying (2.2)

α(X k ) = X k for all k, and α(µx) ∈ α(µ)X for all µ ∈ X k and x ∈ X.

Define αk := α|X k . Then {αk } is an automorphism α of T . The inverse is also an automorphism of T , and also satisfies (2.2).

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Proof. Since α is a bijection and X k is a finite set, α(X k ) = X k implies that αk is a bijection. We need to show that (2.1) is satisfied. Suppose µ ∈ X k and ν ∈ X k+1 . If {µ, ν} is an edge then ν = µx for some x ∈ X and α(ν) = αk+1 (µx) ∈ αk (µ)X, so that {α(µ), α(ν)} is an edge. Now suppose {α(µ), α(ν)} is an edge, say α(ν) = α(µ)y for y ∈ X. Since ν ∈ X k+1 for k ≥ 0 we can write ν = ηx for some η ∈ X k and x ∈ X. Then αk+1 (ν) ∈ α(η)X, α(η) = α(µ), and η = µ. So ν = µx and {µ, ν} is an edge. Thus we have (2.1), and {αk } is an automorphism of T . For the assertion about α−1 , note that (2.1) implies that this inverse is a graph isomorphism in the usual sense, and hence satisfies the weaker condition (2.2) by the comment before the lemma. Suppose again that X is a finite set. An automaton over X is a finite set A together with a map (a, x) 7→ (a · x, a|x ) from A × X to X × A such that, for each fixed a ∈ A, x 7→ a · x is a bijection. For µ ∈ X k and a ∈ A we define a|µ inductively by a|µx = (a|µ )|x . For k ≥ 0 we define fa,k : X k → X k by fa,k (µ) = (a · µ1 )(a|µ1 · µ2 ) · · · (a|µ1 ···µk−1 · µk )

for µ = µ1 · · · µk ∈ X k . Then we have

fa,k+1 (µx) = (a · µ1 )(a|µ1 · µ2 ) · · · (a|µ1 ···µk−1 · µk )(a|µ · x) = fa,k (µ)(a|µ · x) ∈ fa,k (µ)X

from which it follows that fa := {fa,k } is an automorphism of TX for a ∈ A. The set of automorphisms of TX form a group Aut TX . We let GA be the subgroup of Aut TX generated by {fa : a ∈ A}. Recall that a faithful action of a group G on X ∗ is self-similar if for all g ∈ G, x ∈ X there exist y ∈ X, h ∈ G such that g · (xw) = y(h · w) for all w ∈ X ∗ .

(2.3)

Proposition 2.2. There is a faithful self-similar action of GA on TX such that fa · µ := fa (µ) for a ∈ A and µ ∈ X ∗ . Proof. We first show that each fa satisfies condition (2.3), then that each fa−1 does, and then that the composition of two automorphisms satisfying (2.3) also satisfies (2.3). First consider fa for a ∈ A. For x ∈ X and µ ∈ X k fa (xµ) = fa,k+1 (xµ) = (a · x)(a|x · µ1 )(a|xµ1 · µ2 ) · · · (a|xµ1 ···µk−1 · µk ) = (a · x)(a|x · µ1 )((a|x )|µ1 · µ2 ) · · · ((a|x )|µ1 ···µk−1 · µk )

= (a · x)fa|x (µ1 · · · µk )

and hence we can take y = a · x and h = fa|x in (2.3). Now consider the automorphism fa−1 for a ∈ A. fa−1 (xµ) = yν ⇔ xµ = fa (yν)

⇔ xµ = (a · y)fa|y (ν)

⇔ x = a · y and µ = fa|y (ν)

⇔ y = fa−1 (x) and ν = fa|−1y (ν)

⇔ y = fa−1 (x) and ν = fa|−1−1 (ν). fa

So fa−1 (xµ) =

fa−1 (x)fa|−1−1 (ν), f (x) a

(x)

and we can take y = fa−1 (x) and h = fa|−1−1 fa

in (2.3). (x)

SELF-SIMILAR ACTIONS OF GROUPOIDS ON GRAPHS b

5

(y,y) (x,x)

(x,y)

(x,x)

e (y,y)

a

(y,x)

Figure 1. The Moore diagram for the generators of the basilica group. Finally, suppose gi ∈ GA and for each x ∈ X we have yi,x ∈ X and hi,x ∈ GA such that gi (xµ) = yi,x hi,x (µ) for all µ.

Then for each µ ∈ X ∗

g1 g2 (xµ) = g1 (y2,x h2,x (µ)) = y1,y2,x h1,y2,x (h2,x (µ)) = y1,y2,x (h1,y2,x h2,x )(µ).

Thus y = y1,y2,x and h = h1,y2,x h2,x satisfy the property in (2.3) for g1 g2 . Finally, since every element of GA = hfa : a ∈ Ai is a product of elements of the form fa and fb−1 for a, b ∈ A, we can for every g ∈ GA construct elements y ∈ X and h ∈ GA with the required properties. Example 2.3. Suppose X = {x, y}, and an automaton has alphabet A = {a, b, e}. The actions of a, b ∈ A on X = {x, y} satisfy a · x = y, a|x = b, a · y = x, a|y = e, b · x = x, b|x = a, b · y = y, and b|y = e; e ∈ A acts and restricts trivially, so e · x = x and e|x = e, and similarly on y. This is typically presented as a recursive definition on X ∗ by (2.4)

a · (xw) = y(b · w)

a · (yw) = xw

and

b · (yw) = yw for w ∈ X ∗ .

b · (xw) = x(a · w)

We illustrate an automaton by drawing a Moore diagram. In a Moore diagram the arrow g

(x,y)

h

means that g · x = y and g|x = h. Thus, for example, the automaton representing the generators A for the basilica group is illustrated by the Moore diagram in Figure 1. The identity e is often not included in the alphabet A, but is implicit in the recursive definition [17, (2.6)]. 3. Self-similar groupoids and automata over graphs We now generalise the construction of §2, replacing the set X by the set of edges E 1 in a finite directed graph E. Suppose E = (E 0 , E 1 , r, s) is a directed graph with vertex set E 0 , edge set E 1 , and range and source maps r, s : E 1 → E 0 . Write E k = {µ = µ1 · · · µk : µi ∈ E 1 , s(µi ) = r(µi+1 )}

0 ∗ for F the kset of paths of length k in E, keep E for the set of vertices, and define E = k≥0 E . We recover the classical situation of the previous section by taking E to be the

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TE

E 4 1

v

3 2

v

w

1

w 11

2 12

23

3 24

31

4 32

41

42

Figure 2. The forest of trees TE for the graph E graph ({∗}, X, r, s) in which r(x) = r(y) = s(x) = s(y) = ∗ for all x, y ∈ X = E 1 and E ∗ = X ∗. The analogue of the tree TX in our situation is the (undirected) graph TE with vertex set T 0 = E ∗ and edge set T 1 = {µ, µe} : µ ∈ E ∗ , e ∈ E 1 , and s(µ) = r(e) .

The subgraph vE ∗ = {µ ∈ E ∗ : r(µ) = v} is a rooted tree with root v ∈ E 0 , and F TE = v∈E 0 vE ∗ is a disjoint union of trees, or forest. An example of a graph E and the corresponding forest TE is given in Figure 2. We will be using range and source maps in the context of both graphs and, later, automata. If there is any room for confusion or if we simply want to be clear for later reference, we will use the notation rE and sE for the range and source maps associated to E. Difficulties in defining restrictions consistently in this new context mean that we need to deal with partial isomorphisms rather than automorphisms. Partial isomorphisms of TE are only defined on a subtree of TE but are still isomorphisms of their domain onto their range. Definition 3.1. Suppose E = (E 0 , E 1 , r, s) is a directed graph. A partial isomorphism of TE consists of two vertices v, w ∈ E 0 and a bijection g : vE ∗ → wE ∗ such that (3.1)

g|vE k is a bijection onto wE k for k ∈ N, and g(µe) ∈ g(µ)E 1 for all µe ∈ vE ∗ .

For v ∈ E 0 , we write idv : vE ∗ → vE ∗ for the partial isomorphism given by idv (µ) = µ for all µ ∈ vE ∗ . We denote the set of all partial isomorphisms of TE by PIso(E ∗ ). We define domain and codomain maps d, c : PIso(E ∗ ) → E 0 so that g : d(g)E ∗ → c(g)E ∗; thus in (3.1) we have d(g) := v and c(g) := w. Because we work with partial isomorphisms instead of automorphisms, we work with groupoids rather than groups. A groupoid differs from a group in two main ways: the product in a groupoid is only partially defined, and a groupoid typically has more than one unit. A groupoid is a small category with inverses. Thus a groupoid consists of a set G0 of objects, a set G of morphisms, two functions c, d : G → G0 , and a partially defined product (g, h) 7→ gh from G2 := {(g, h) : d(g) = c(h)} to G such that (G, G0 , c, d) is a category and such that each g ∈ G has an inverse g −1. (We then typically write G to denote the groupoid, and call G0 the unit space of the groupoid. If |G0 | = 1, then G is


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simply a group in the usual sense.) There are many formulations of this definition; we have chosen this one because it emphasises the objects and morphisms as distinct sets, and neatly summarises the axioms. The next result describes an important example, and its proof illustrates the efficiency of the definition. Proposition 3.2. Suppose that E = (E 0 , E 1 , r, s) is a directed graph with associated forest TE , and resume the notation of Definition 3.1. Then (PIso(E ∗ ), E 0 , c, d) is a groupoid in which: the product is given by composition of functions, the identity isomorphism at v ∈ E 0 is idv : vE ∗ → vE ∗ , and the inverse of g ∈ PIso(E ∗ ) is the inverse of the function g : d(g)E ∗ → c(g)E ∗ . Proof. For g, h ∈ PIso(E ∗ )2 , the composition gh of h : d(h)E ∗ → c(h)E ∗ with g : d(g)E ∗ = c(h)E ∗ → c(g)E ∗ is a bijection of d(h)E ∗ onto c(g)E ∗ which maps each d(h)E k onto c(g)E k . To see that gh has the second property in (3.1), we take µe ∈ d(h)E ∗ . Since h satisfies (3.1), there exists f ∈ E 1 such that h(µe) = h(µ)f , and then (gh)(µe) = g(h(µe)) = g(h(µ)f )

belongs to g(h(µ))E 1 = (gh)(µ)E 1 because g satisfies (3.1). So gh is a partial isomorphism with d(gh) = d(h) and c(gh) = c(g). Associativity of the multiplication follows from the associativity of composition. Thus (PIso(E ∗ ), E 0 , c, d) is a category. The usual properties of composition also imply that idc(g) g = g = g idd(g) . Each g : d(g)E ∗ → c(g)E ∗ is a bijection, and hence has a set-theoretic inverse g −1 : c(g)E ∗ → d(g)E ∗. That this inverse is indeed a partial isomorphism follows from the last assertion in Lemma 2.1 (or rather from its extension to partial isomorphisms). Suppose that E is a directed graph and G is a groupoid with unit space E 0 . An action of G on the path space E ∗ is a (unit-preserving) groupoid homomorphism φ : G → PIso(E ∗ ); the action is faithful if φ is one-to-one. If the homomorphism is fixed, we usually write g · µ for φg (µ). This applies in particular when G arises as a subgroupoid of PIso(E ∗ ). Definition 3.3. Suppose E = (E 0 , E 1 , r, s) is a directed graph and G is a groupoid with unit space E 0 which acts faithfully on TE . The action is self-similar if for every g ∈ G and e ∈ d(g)E 1, there exists h ∈ G such that (3.2)

g · (eµ) = (g · e)(h · µ) for all µ ∈ s(e)E ∗ .

Since the action is faithful, there is then exactly one such h ∈ G, and we write g|e := h. Definition (3.3) has some immediate and important consequences. Lemma 3.4. Suppose E = (E 0 , E 1 , r, s) is a directed graph and G is a groupoid with unit space E 0 acting self-similarly on TE . Then for g, h ∈ G with d(h) = c(g) and e ∈ d(g)E 1 , we have (1) d(g|e) = s(e) and c(g|e ) = s(g · e), (2) r(g · e) = g · r(e) and s(g · e) = g|e · s(e), (3) if g = idr(e) , then g|e = ids(e) , and (4) (hg)|e = (h|g·e )(g|e ). Proof. For every µ ∈ s(e)E ∗ , g · (eµ) = (g · e)(g|e · µ) is a path in E. This says, first, that µ is in the domain d(g|e)E ∗ of g|e , so that d(g|e) = s(e), and, second, that c(g|e ) = r(g|e · µ) = s(g · e). This gives (1).

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For (2), we observe that µ 7→ g · µ is an isomorphism of the tree d(g)E ∗ onto c(g)E ∗, and in particular we have g · d(g) = c(g). Thus r(g · e) = c(g) = g · d(g) = g · r(e). On the other hand, g|e is an isomorphism of d(g|e )E ∗ = s(e)E ∗ onto c(g|e )E ∗ = s(g · e)E ∗ . So g|e · s(e) is s(g · e). If g = idr(e) , then g · (eµ) = eµ = e(idr(µ) ·µ) = e(ids(e) ·µ), and the uniqueness of restrictions gives (3). For (4), we take µ ∈ d(g)E ∗ and compute: ((hg) · e)((hg)|e · µ) = (hg) · (eµ) = h · ((g · e)(g|e · µ)) = (h · (g · e))(h|g·e · (g|e · µ))

= ((hg) · e)((h|g·e g|e ) · µ).

Thus we have (hg)|e · µ = (h|g·e g|e ) · µ for all µ ∈ d(g)E ∗, and (4) follows because the action is faithful. Remark 3.5. Lemma 3.4(2) implies that the source map may not be equivariant: s(g · e) 6= g · s(e) in general. Indeed, g · s(e) will often not make sense: g maps d(g)E ∗ onto c(g)E ∗, and s(e) is not in d(g)E ∗ unless s(e) = d(g). This non-equivariance of the source map is one of the main points of difference between our work and that of Exel–Pardo [8] and Bedos–Kaliszewski–Quigg [2]; see Appendix A for further details. Suppose µ = µ1 · · · µm ∈ d(g)E m and ν ∈ s(µ)E ∗ . Then µν ∈ d(g)E ∗ and

g · (µν) = (g · µ1 )(g|µ1 · µ2 ) · · · (· · · (g|µ1 )|µ2 ) · · · )|µm−1 · µm )((· · · (g|µ1 )|µ2 ) · · · )|µm · ν) = (g · µ)((· · · (g|µ1 )|µ2 ) · · · )|µm · ν);

thus with g|µ := (· · · (g|µ1 )|µ2 ) · · · )|µm we can generalise (3.2) to

(3.3)

g · (µν) = (g · µ)(g|µ · ν) for all µ ∈ d(g)E ∗, ν ∈ s(µ)E ∗ .

By iterating parts (3) and (4) of Lemma 3.4 we arrive at the following more general versions: Proposition 3.6. Suppose E = (E 0 , E 1 , r, s) is a directed graph and G is a groupoid with unit space E 0 acting self-similarly on TE . Then for all g, h ∈ G, µ ∈ d(g)E ∗ , and ν ∈ s(µ)E ∗ we have: (1) g|µν = (g|µ )|ν , (2) idr(µ) |µ = ids(µ) , (3) (hg)|µ = h|g·µ g|µ , and (4) g −1|µ = (g|g−1·µ )−1 . Proof. The first three of these are straightforward. For the last, observe that on the one hand, we have gg −1 = idr(µ) , so (gg −1)|µ = ids(µ) , and on the other we have (gg −1)|µ = (g|g−1·µ )(g −1|µ ). Now multiplying ids(µ) = (g|g−1·µ )(g −1 |µ ) on the left by (g|g−1·µ )−1 gives the result. We now generalise the process of constructing faithful self-similar actions from automata. The switch to path spaces requires that both the action and restriction maps interact with the graph structure. We begin by defining automata in this more general context. Since the groupoid PIso(E ∗ ) has local identities idv associated to each vertex v, we may need to add some of the vertices to the alphabet A of the automaton. If our aim were only to ensure that the set A is closed under taking restrictions then we may only need


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to add some of the vertices — see Example 3.10, where we would need to add v to A but not w. However, we want groupoids generated from automata associated to E to be subgroupoids of PIso(E ∗ ). Since the standard definition of a subgroupoid involves having the same unit space as the ambient groupoid, we assume that E 0 ⊂ A.

Definition 3.7. An automaton over E = (E 0 , E 1 , rE , sE ) is a finite set A containing E 0 together with functions rA , sA : A → E 0 such that rA (v) = v = sA (v) if v ∈ E 0 ⊂ A, and a function (3.4)

A sA×rE E 1 ∋ (a, e) 7→ (a · e, a|e ) ∈ E 1

s E × rA

A

such that: (A1) for every a ∈ A, e 7→ a · e is a bijection of sA (a)E 1 onto rA (a)E 1 ; (A2) sA (a|e ) = sE (e) for all (a, e) ∈ A sA×rE E 1 ; (A3) rE (e) · e = e and rE (e)|e = sE (e) for all e ∈ E 1 .

Since sE (v) = rE (v) = sA (v) = rA (v) = v for all v ∈ E 0 , the range and source maps are consistent whenever they both make sense. So there should be no confusion in using s, r for both the graph and the automaton. However, sometimes it is convenient to distinguish between the two for clarity as in Definition 3.7, in which case we write sE , rE and sA , rA for the range and source maps in E and A respectively. We can use the property (A2) to extend restriction to paths by defining a|µ = (· · · ((a|µ1 )|µ2 )|µ3 · · · )|µk .

The point is that sA (a|µ1 ) = sE (µ1 ) = rE (µ2 ), for example, and hence (a|µ1 )|µ2 makes sense. Our next result says that we can extend the action of elements of the set A to partial isomorphisms on E ∗ . Proposition 3.8. Suppose that E is a directed graph and A is an automaton over E. We recursively define maps fa,k : s(a)E k → r(a)E k for a ∈ A and k ∈ N by fa,1 (e) = a · e and (3.5)

fa,k+1 (eµ) = (a · e)fa|e ,k (µ) for eµ ∈ sA (a)E k+1 .

Then for every a ∈ A, fa = {fa,k } is a partial isomorphism of s(a)E ∗ onto r(a)E ∗ so that d(fa ) = s(a) and c(fa ) = r(a); for a = v ∈ A ∩ E 0 , we have fa = idv : vE ∗ → vE ∗ .

Proof. We prove by induction on k that the set {fa,k : a ∈ A} consists of bijections fa,k of s(a)E k onto r(a)E k . The base case k = 1 is (A1) in the definition of automaton. So we suppose the claim is true for k ≥ 1. For eµ ∈ s(a)E k+1 , then s(a|e ) = s(e) = r(µ), so the right-hand side of (3.5) makes sense and belongs to r(a)E k+1. To see that fa,k+1 is one-to-one, suppose that fa,k+1 (eµ) = fa,k+1 (f ν). Then we have a · e = a · f , which implies e = f by (A1), and then fa|e ,k (µ) = fa|f ,k (ν) = fa|e ,k (ν) =⇒ µ = ν, and eµ = f ν. To see that fa,k+1 is onto, take f ν ∈ r(a)E k+1 . Then (A1) gives e ∈ E 1 such that a · e = f , and surjectivity of fa|e ,k gives µ such that fa|e ,k (µ) = ν. Then fa,k+1 (eµ) = f ν, and fa,k+1 is onto. We now have to prove that the maps fa,k satisfy the second condition in (3.1), which we again do simultaneously for all a by induction on k. Let a ∈ A, µ ∈ s(a)E k , e ∈ s(µ)E 1 , and we want fa,k+1 (µe) ∈ fa,k (µ)E 1 . For k = 1, we have |µ| = 1 and µ = µ1 ∈ E 1 . Thus (3.5) gives fa,2 (µe) = (a · µ1 )fa|µ1 ,1 (e) = (a · µ1 )(a|µ1 · e) ∈ fa,1 (µ1 )E 1 .

10


E 4 1

v

3

w

2 Figure 3. Graph for Example 3.10 We now suppose that what we want is true for {fa,k : a ∈ A} for some k ≥ 1, and take µe ∈ s(a)E k+2 . Then since |µ| = k + 1, we can factor µe = µ1 µ′ e, and then fa,k+2 (µe) = (a · µ1 )fa|µ1 ,k+1(µ′ e) ∈ (a · µ1 )fa|µ1 ,k (µ′ )E 1 = fa,k+1 (µ)E 1 ,

as required. Since r(v) = v = s(v), v ·e = e and v|e = v for all e ∈ vE ∗ , a simple induction argument shows that fv is the identity on vE ∗ . Theorem 3.9. Suppose that E is a directed graph and A is an automaton over E. For a ∈ A, let fa be the partial isomorphism of TE described in Proposition 3.8, and let GA be the subgroupoid of PIso(E ∗ ) generated by {fa : a ∈ A} (which by convention includes the identity morphisms {idv : v ∈ E 0 }). Then GA acts faithfully on the path space E ∗ , and this action is self-similar. This is the point where it really helps to have worked carefully through the classical case in which |E 0 | = 1 and E 1 is just a finite set X. Notice that GA acts faithfully because it is by definition a subgroupoid of PIso(E ∗ ). Proof of Theorem 3.9. The inverses fa−1 : r(a)E ∗ → s(a)E ∗ of the bijections fa : s(a)E ∗ → r(a)E ∗ are also partial isomorphisms (see Proposition 3.2; the crucial argument is that of Lemma 2.1, which shows that fa−1 satisfies (3.2)). Then every element of GA is a finite product of partial isomorphisms fa and fb−1 in which the domains and codomains match up. (So, for example, fa fb−1 where s(a) = c(fb−1) = s(b).) Thus it suffices for us to show first, that if a partial automorphism g ∈ PIso(E ∗ ) has restrictions g|e for all e ∈ d(g)E 1, then so does g −1; and second, that if gi ∈ PIso(E ∗ ) have d(g1 ) = c(g2 ), and gi have restrictions gi |e for all e ∈ d(gi )E 1 , then so does g1 g2 . But for these we just need to look at the second and third paragraphs in the proof of Proposition 2.2 to see that they carry over verbatim. So GA acts self-similarly, as claimed. Example 3.10. We consider the graph E in Figure 3, so that E 0 = {v, w}, E 1 = {1, 2, 3, 4}, v = sE (1) = rE (1) = rE (2) = sE (3) = sE (4) and w = sE (2) = rE (3) = rE (4). We take A = {a, b} ∪ E 0 , and define rA , sA : A → E 0 by sA (a) = v = rA (b) and sA (b) = w = rA (a). We define (3.6)

a · 1 = 4, a|1 = v; a · 2 = 3, a|2 = b;

b · 3 = 1, b|3 = v;

b · 4 = 2, b|4 = a.

(The restrictions on E 0 ⊂ A are specified in (A3).) We verify that e 7→ a · e is a bijection of sA (a)E 1 = vE 1 = {1, 2} onto rA (a)E 1 = wE 1 = {3, 4}, and correspondingly for b. For


11

(A2), we check sA (a|1 ) = sA (v) = v = sE (1) sA (a|2 ) = sA (b) = w = sE (2) sA (b|3 ) = sA (v) = v = sE (3) sA (b|4 ) = sA (a) = v = sE (4) The data (3.6) is often presented as (3.7)

a · 1µ = 4µ

a · 2ν = 3(b · ν)

rA (a|1 ) = rA (v) = v = sE (a · 1) rA (a|2 ) = rA (b) = v = sE (a · 2) rA (b|3 ) = rA (v) = v = sE (b · 3)

rA (b|4 ) = rA (w) = w = sE (b · 4). b · 3µ = 1µ

b · 4µ = 2(a · µ),

from which we are meant to deduce (3.6). To see how this deduction is meant to work, we observe that the formulas in (3.7) allow us to define partial isomorphisms fa and fb recursively, using the construction that was formalised in Proposition 3.8. So a and b have to be in the alphabet; we take A = {a, b} ∪ E 0 . Consider the action of a ∈ A. By assuming 1µ ∈ E ∗ we are assuming µ ∈ sE (1)E ∗ = vE ∗ . Since a · 1µ = 4µ for all µ ∈ vE ∗ , it follows from Proposition 3.8 that a · 1 = 4 and fa|1 (µ) = µ for all µ ∈ vE ∗ . So whatever a|1 is, it must satisfy fa|1 = idv ; since the partial isomorphisms fa , fb , idv and idw all act differently on the edge set E 1 , they are distinct partial isomorphisms, and the only choice for a|1 in our current alphabet is a|1 := v. Similarly, we define a|2 = b, b|3 = v and b|4 = a. We don’t need to specify restrictions for v and w, because the condition r(e)|e = s(e) forces v|1 = v, v|2 = w and w|3 = w|4 = v. Notice that we made choices in the last paragraph: we could have added extra elements to our alphabet. 4. The Toeplitz algebra of a self-similar groupoid action Suppose that G is a (discrete) groupoid. Then the groupoid elements g ∈ G give point masses ig in Cc (G), and Cc (G) = span{ig : g ∈ G}. For g, h ∈ G, the involution and product described in [6, Proposition 3.11] are determined by ( igh if d(g) = c(h) i∗g = ig−1 and ig ∗ ih = 0 otherwise. A function U : G → B(H) is a unitary representation of G if • for v ∈ G0 , Uv is the orthogonal projection on a closed subspace of H, • for each g ∈ G, Ug is a partial isometry with initial projection Ud(g) and final projection Uc(g) , and • for g, h ∈ G, we have ( Ugh if d(g) = c(h) (4.1) Ug Uh = 0 otherwise. (The multiplication formula (4.1) applies when g and h are units, and then says that the projections Uv are mutually orthogonal.) We have a similar notion of unitary representation with values in a C ∗ -algebra, and then the map i : g 7→ ig is a unitary representation of G in Cc (G) ⊂ C ∗ (G). We have chosen the name “unitary representation” to emphasise that each Ug is a unitary isomorphism of Ud(g) H onto Uc(g) H.

12


Proposition 4.1. Let G be a groupoid. Then the pair (C ∗ (G), i) is universal for unitary representations of G. Proof. Suppose that U : G → B(H) is a unitary representation. Then extending U linearly gives a ∗-representation π of Cc (G) = span{ig } in B(H) such that U = π ◦ i. Since this representation is bounded for the norm of C ∗ (G) (see [6, page 205]), π extends to a representation πU of C ∗ (G) on H such that πU ◦ i = U.

Example 4.2. For any groupoid G, there is a unitary representation U of G on ℓ2 (G0 ) in which Ug is the matrix unit ec(g),d(g) . The corresponding homomorphism πU : C ∗ (G) → B(ℓ2 (G0 )) takes elements of Cc (G) into finite rank operators, and hence has range in the algebra K(ℓ2 (G0 )) of compact operators. When G0 is finite, πU takes values in MG0 (C), and composing with the normalised trace gives a tracial state on C ∗ (G). This is the analogue for groupoids of the trace τ1 on group algebras used in [17] (see page 6648). If G is not transitive, then the range of πU will be a proper subalgebra of MG0 (C), and the normalised traces on simple summands will give other tracial states on C ∗ (G). Suppose that E is a finite directed graph and (G, E) is a self-similar groupoid action. Consider the graph bimodule X(E) of [9], using the conventions of [23, Chapter 8]. The inclusion of C(E 0 ) = Cc (G0 ) in Cc (G) gives a unital homomorphism of C(E 0 ) into C ∗ (G), which makes C ∗ (G) into a right-Hilbert C(E 0 )-C ∗(G) bimodule. We now form the internal tensor product M := X(E) ⊗C(E 0 ) C ∗ (G), which is a right Hilbert C ∗ (G)-module with inner product given on elementary tensors by hx ⊗ a, y ⊗ bi = h hy, xia, bi = a∗ hx, yib.

Forming the internal tensor product involves taking a completion, which in turn involves modding out by elements with norm 0. In this process, the tensor product becomes balanced over C(E 0 ) (see the foot of [24, page 34], for example). However, the bimodule X(E) is spanned by the point masses {e : e ∈ E 1 }, and the elements e ⊗ 1 form a Parseval frame for M. In particular, every m ∈ M is a finite sum X e ⊗ he ⊗ 1, mi (4.2) m= e∈E 1

of elementary tensors; we refer to (4.2) as the reconstruction formula. Thus M coincides with the algebraic tensor product X(E) ⊙C(E 0 ) C ∗ (G), and no completion is required.

Proposition 4.3. Suppose that (G, E) is a self-similar groupoid action. Then there is a unitary representation W : G → L(M) such that, for g ∈ G, e ∈ E 1 and a ∈ C ∗ (G), ( (g · e) ⊗ ig|e a if d(g) = r(e) (4.3) Wg (e ⊗ a) = 0 otherwise. Proof. Fix g ∈ G. Then, motivated by the reconstruction formula (4.2), we define Wg : M → M by X Wg m = (g · f ) ⊗ (ig|f hf ⊗ 1, mi). f ∈E 1 , d(g)=r(f )

Now taking m = e ⊗ a and considering only the edges f with r(f ) = d(g) (that is, the only edges which appear in the sum), we have ig|f hf ⊗ 1, e ⊗ ai = ig|f hhe, f i1, ai = ig|f hf, eia = δe,f ig|e is(e) a,


13

which is ig|e a because g|e s(e) = g|e in G. Thus Wg satisfies (4.3). The original formula shows that Wg is right C ∗ (G)-linear, and since the sum is finite, an application of the Cauchy-Schwarz inequality shows that Wg is norm-bounded. We next show that Wg is adjointable with Wg∗ = Wg−1 . We take e, f ∈ E 1 and a, b ∈ C ∗ (G), and suppose first that hWg (e ⊗ a), f ⊗ bi = 6 0. The formula (4.3) shows that d(g) = r(e), and then 0 6= hWg (e ⊗ a), f ⊗ bi = h hf, g · ei · ig|e a, bi = a∗ i∗g|e hg · e, f ib.

(4.4)

We deduce that hg · e, f i = 6 0, and hence f = g · e. Thus

r(f ) = r(g · e) = g · r(e) = g · d(g) = c(g) = s(g −1 ),

and we have (4.5)

he ⊗ a, Wg−1 (f ⊗ b)i = h hg −1 · f, ei · a, ig−1 |f bi = a∗ he, g −1 · f iig−1 |f b.

Then hg · e, f i = is(f ) and he, g −1 · f i = is(e) , so

is(g·e) = ig−1 |g·e is(g·e) i∗g|e hg · e, f i = i∗g|e is(f ) = ig|−1 e = ig−1 |g·e = is(e) ig−1 |g·e = he, g −1 · f iig−1 |f .

6 0, Thus we have (4.4)=(4.5) when hWg (e ⊗ a), f ⊗ bi = 6 0. When he ⊗ a, Wg−1 (f ⊗ b)i = we can apply the preceding argument to he ⊗ a, Wg−1 (f ⊗ b)i∗ = hWg−1 (f ⊗ b), e ⊗ ai, and deduce that (4.5)=(4.4). Thus we have equality in all cases, and Wg is adjointable with Wg∗ = Wg−1 . Finally, a short calculation using Proposition 3.6(3) shows that W is a unitary representation of G. The universal property of (C ∗ (G), i) (Proposition 4.1) gives us a unital homomorphism πW : C ∗ (G) → L(M) such that πW (ig ) = Wg for g ∈ G. We use this homomorphism to define a left action of C ∗ (G) on the right-Hilbert C ∗ (G) module M = X(E) ⊗C(E 0 ) C ∗ G), and M thus becomes a Hilbert bimodule over C ∗ (G) (alternatively known as a C ∗ -correspondence over C ∗ (G)). The Toeplitz algebra of the self-similar groupoid action (G, E) is then by definition the Toeplitz algebra T (G, E) := T (M) of the bimodule M. As in [9, §1], we view it as the C ∗ -algebra generated by a universal Toeplitz representation (iM , iC ∗ (G) ) : M → T (M). We now give a presentation of the Toeplitz algebra T (G, E). Proposition 4.4. Let E be a finite graph without sources and (G, E) a self-similar groupoid action. We define u : G → T (G, E) = T (M), p : E 0 → T (M) and s : E 1 → T (M) by u = iC ∗ (G) ◦ i, pv = iC ∗ (G) (iv ), and se = iM (e ⊗ 1). Then (1) u is a unitary representation of G with uv = pv for v ∈P E0; (2) (p, s) is a Toeplitz-Cuntz-Krieger family in T (M), and v∈E 0 pv is an identity for T (M);

14


(3) for g ∈ G and e ∈ E 1 , we have ( sg·e ug|e u g se = 0

if d(g) = r(e) otherwise;

(4) for g ∈ G and v ∈ E 0 , we have ( pg·v ug u g pv = 0

if d(g) = v otherwise.

The C ∗ -algebra T (M) is generated by {ug }∪{pv }∪{se }, and (T (M), (u, p, s)) is universal for families U, P , S satisfying the upper-case analogues of (1–4). Proof. Part (1) holds because i : G → C ∗ (G) is a unitary representation. For part (2), we begin by observing that the point masses {v : v ∈ E 0 } are mutually 0 orthogonal projections in C(E P ), and hence the {pv } are mutually orthogonal projections in T (M). In particular q = v∈E 0 pv is also a projection. Next we note that the elements {e ⊗ ig : e ∈ E 1 , g ∈ G, s(e) = c(g)}

span a dense subspace of M, and that for each such element we have X q(e ⊗ ig ) = (v · e) ⊗ ig = (s(e) · e) ⊗ ig = e ⊗ ig . v∈E 0

Thus the bimodule M is essential in the sense of [9], and it follows that q is an identity for T (M). (See the discussion in the middle of [9, page 178].) We next verify the ToeplitzCuntz-Krieger relations. We take e, f ∈ E 1 and compute s∗e sf = iMe (e ⊗ 1)∗ iMe (f ⊗ 1) = iC ∗ (G) he ⊗ 1, f ⊗ 1i = δe,f iC ∗ (G) (s(e) · 1) = δe,f iC ∗ (G) (is(e) ) = δe,f ps(e) ;

this proves that s∗s se = ps(e) , and that the elements {se s∗e : e ∈ E 1 } are mutually orthogonal projections. We calculate (4.6)

pv se = iC ∗ (G) (iv )iM (e ⊗ 1) = iM (Wv (e ⊗ 1)) = iM ((v · e) ⊗ iv|e 1) = δv,r(e) iM (e ⊗ is(e) ) = δv,r(e) iMe (e ⊗ 1) = δv,r(e) se ;

thus pr(e) ≥ se s∗e , and, since the {se s∗e } are mutually orthogonal, pv ≥ (p, s) is a Toeplitz-Cuntz-Krieger E-family. To establish part (3), we compute

P

e∈vE 0

se s∗e . Thus

ug se = iC ∗ (G) (ig )iM (e ⊗ 1) = iM (ig · (e ⊗ 1))

= δd(g),r(e) iM ((g · e) ⊗ ig|e ) = δd(g),r(e) iM (((g · e) ⊗ 1) · ig|e ) = δd(g),r(e) iM ((g · e) ⊗ 1)iC ∗ (G) (ig|e ) = δd(g),r(e) sg·e ug|e .

A similar computation gives (4). Since T (M) is generated by the images of iM = span{e ⊗ ig } and iC ∗ (G) = span{ig }, the tg , pv and se generate. It remains to verify the universal property, so we take Ug , Pv and Se in a C ∗ -algebra B satisfying the relations (1–4).


15

The unitary representation U of G induces a homomorphism πU : C ∗ (G)P → B such P that πU (ig ) = Ug . Since v∈E 0 iv is the identity in C ∗ (G) and (2) says that v∈E 0 Pv is the identity in B, πU is unital. Define ψ : M → B by X Se πU (he ⊗ 1, mi). ψ(m) = e∈E 1

We will prove that (ψ, πU ) is a Toeplitz representation of M. An inspection of the formula for ψ(m) shows that ψ(m · a) = ψ(m)πU (a). Next, for m, n ∈ M we have X πU (he ⊗ 1, mi∗ Se∗ Sf πU (hf ⊗ 1, ni). ψ(m)∗ ψ(n) = e,f ∈E 1

The summands with e 6= f vanish, and Thus

Se∗ Se πU (he ⊗ 1, ni) = Ps(e) πU (he ⊗ 1, ni) = πU (is(e) he ⊗ 1, ni) = πU (he ⊗ 1, ni). ψ(m)∗ ψ(n) =

X

e∈E 1

= πU

πU he ⊗ 1, mi∗ he ⊗ 1, ni

X

e∈E 1

= πU

D

m,

hm, e ⊗ 1ihe ⊗ 1, ni X

e∈E 1

(e ⊗ 1)he ⊗ 1, ni

E

,

which by the reconstruction formula is πU (hm, ni). Next we have to verify that ψ(b · m) = πU (b)ψ(m) for b ∈ C ∗ (G) and m ∈ M. For this we take b = ig , m = f ⊗ a and compute both sides. On one hand, we have ψ(ig (f ⊗ a)) = ψ(Wg (f ⊗ a)),

which is 0 unless d(g) = r(f ). If d(g) = r(f ), then

ψ(ig (f ⊗ a)) = ψ((g · f ) ⊗ ig|f a) X Se he ⊗ 1, gf ⊗ ig|f ai = e∈E 1

=

X

e∈E 1

Se πU

hg · f, ei, ig|f a

= sg·f πU (ig|f a). On the other hand, we have πU (ig )ψ(f ⊗ a) =

X

e∈E 1

Ug Se πU he ⊗ 1, f ⊗ ai = Ug Sf πU (a).

The formula in (3) says that this too is 0 unless d(g) = r(f ), and then gives πU (ig )ψ(m) = Sg·f Ug|f πU (a) = Sg·f πU (ig|f a), which is ψ(ig (f ⊗ a)). Now the universal property of T (M) give a homomorphism ψ × πU : T (M) → B such that ψ = (ψ × πU ) ◦ iM and πU = (ψ × πU ) ◦ iC ∗ (G) . To finish, we need to check that

16


ψ × πU maps (u, p, s) to (U, P, S). This is straightforward for u and p. For f ∈ E 1 , we have (ψ × πU )(sf ) = (ψ × πU )(iM (f ⊗ 1)) = ψ(f ⊗ 1) X Se πU (he ⊗ 1, f ⊗ 1i), = e∈E 1

which collapses to Sf πU (is(f ) ) = Sf Ps(f ) = Sf .

Proposition 4.5. Suppose (G, E) is a faithful self-similar groupoid action and take (u, p, s) as in Proposition 4.4. Then T (G, E) = span{sµ ug s∗ν : µ, ν ∈ E ∗ , g ∈ G and s(µ) = g · s(ν)}. The condition s(µ) = g · s(ν) is just there to exclude elements which are certainly 0: sµ ug s∗ν 6= 0 =⇒ sµ ps(µ) ug ps(ν) s∗ν 6= 0 =⇒ ps(µ) ug ps(ν) 6= 0 =⇒ ps(µ) pg·s(ν) 6= 0.

The closed linear span on the right-hand side contains all the generators, and hence the result follows from the following lemma, which is very similar to [17, Lemma 3.4]. Lemma 4.6. Suppose κ, λ, µ, ν ∈ E ∗ and g, h ∈ S satisfy s(κ) = g·s(λ) and s(µ) = h·s(ν) we have  ∗  if µ = λµ′ sκ(g·µ′ ) ug|µ′ h sν (4.7) (sκ ug s∗λ )(sµ uh s∗ν ) = sκ ugh|h−1 ·λ′ s∗ν(h−1 ·λ′ ) if λ = µλ′   0 otherwise.

Proposition 4.7. Suppose (G, E) is a self-similar groupoid action and let (u, p, s) be the universal representation from Proposition 4.4. Then O(G, E) is the quotient of T (G, E) by the ideal generated by o n X pv − se s∗e : v ∈ E 0 . {e∈vE 1 }

Proof. Let φ : C ∗ (G) → L(M) be the homomorphism implementing the left action, and let (ψ, π) be a Toeplitz representation of M. Then the reconstruction formula implies that X Θe⊗1,a∗ (e⊗1) for all a ∈ C ∗ (G). φ(a) = e∈E 1

∗

Thus for every a ∈ C (G) we have X X se s∗e π(a). ψ(e ⊗ 1)ψ(a∗ · (e ⊗ 1))∗ = (4.8) (ψ, π)(1) (φ(a)) = e∈E 1

e∈E 1

If (ψ, π) is Cuntz-Pimsner covariant, then taking a = iv in (4.8) gives X π(pv ) = (ψ, π)(1) (φ(pv )) = ψ(se )ψ(se )∗ . r(e)=v

Conversely, if we have the Cuntz Krieger relation at every v, then for every g ∈ G we have X π(ig ) = π(ic(g) ig ) = ψ(se )ψ(se )∗ π(ig ) r(e)=c(g)


=

X

17

ψ(se )ψ(se )∗ π(ig )

e∈E 1

= (ψ, π)(1) (φ(ig )).

5. An algebraic characterisation of KMS states Suppose that E is a finite graph with no sources and that (G, E) is a self-similar groupoid action over E. There is a strongly continuous gauge action γ : T → Aut T (G, E) such that γz (iC ∗ (G) (a)) = iC ∗ (G) (a) and γz (iM (m)) = ziM (m) for a ∈ C ∗ (G) and m ∈ M [9, Proposition 1.3 (c)]. The gauge action gives rise to a periodic action σ of the real line (a dynamics) by the formula σt = γeit . This dynamics satisfies (5.1)

σt (sµ ug s∗ν ) = eit(|µ|−|ν|) sµ ug s∗ν .

Suppose that σ : R → Aut A is a strongly continuous action on a C ∗ -algebra A. An element a ∈ A is analytic if the function t 7→ σt (a) extends to an entire function on C. For β ∈ R a state φ on A is a KMSβ state for (A, σ) if it satisfies the KMSβ condition (5.2)

φ(ab) = φ(bσiβ (a)),

for all a, b in a set of analytic elements that spans a dense σ-invariant subspace of A. For β 6= 0 all KMSβ states are σ-invariant, see [3, Proposition 5.3.3]; for β = 0 it is customary to make σ-invariance part of the definition, so that the KMS0 states are the σ-invariant traces. We are interested in the KMSβ states of (T (G, E), σ). The spanning elements sµ ug s∗ν are analytic in T (G, E), because the function t ∈ R 7→ eit(|µ|−ν|) has an entire extension to C. Therefore it is enough to check the KMSβ condition on spanning elements. P Since the ideal generated by the elements pv − e∈vE 1 se s∗e is σ-invariant, there is a compatible time evolution (also denoted by σ) on the quotient O(G, E). The KMS states of (O(G, E), σ) are given by KMS states of (T (G, E), σ) that factor through O(G, E). The following alternative characterisation of the KMSβ states of T (G, E) and O(G, E) is motivated by [10, §2] and [17, §4]. Proposition 5.1. Let E be a finite graph with no sources and vertex matrix B, and let ρ(B) be the spectral radius of B. Suppose that (G, E) is a self-similar groupoid action. Let σ : R → Aut T (G, E) be the dynamics given by (5.1). (1) For β < ln ρ(B), there are no KMSβ -states for σ. (2) For β ≥ ln ρ(B), a state φ is a KMSβ -state for σ if and only if φ ◦ iC ∗ (G) is a trace on C ∗ (G) and (5.3)

φ(sµ ug s∗ν ) = δµ,ν δs(µ),c(g) δs(ν),d(g) e−β|µ| φ(ug )

for g ∈ S and µ, ν ∈ E ∗ .

In principle, we should be able to deduce part (2) from the general result in [1, Proposition 3.1], which characterises the KMS states on an arbitrary Toeplitz-Cuntz-Pimsner algebra T (X). However, the description there is in terms of elementary tensors in the tensor powers X ⊗n , and since our module M is itself a tensor product, the tensor powers get quite complicated. So it seems more straightforward to reason directly in terms of our spanning family. Proof. Suppose first that φ is a KMSβ -state of (T (G, E), σ). The Toeplitz-Cuntz-Krieger family (p, s) in T (G, E) induces a homomorphism πp,s : T (E) → T (G, E), and πp,s is

18


equivariant for σ and the periodic dynamics arising from the gauge action on T (E). Thus the composition φ ◦ πp,s is a KMSβ -state on T (E), and [10, Proposition 4.3 (c)] implies that β ≥ ln ρ(B). This proves part (1). For part (2), suppose again that φ is a KMSβ state. Since γ fixes ug and uh , the KMS relation implies that φ(ug uh ) = φ(uh ug ), and hence that φ ◦ iC ∗ (G) is a trace. The second and third delta functions on the right-hand side of (5.3) just reflect that sµ ug s∗ν = 0 unless s(µ) = c(g) and d(g) = s(ν). Since φ is gauge-invariant, φ(sµ ug s∗ν ) = 0 unless |µ| = |ν|, and then using the KMS commutation relation to pull sµ past ug s∗ν gives (5.3). For the reverse implication, suppose that φ ◦ iC ∗ (G) is a trace and that φ satisfies (5.3). To see that φ is a KMSβ state, it suffices to verify the KMS condition (5.4)

φ(bc) = φ(cαiβ (b)) for b = sκ ug s∗λ and c = sµ uh s∗ν .

There is symmetry in (5.4) which we can use to simplify the argument. First, note that both sides vanish unless |κ| + |µ| = |λ| + |ν|. The right-hand side is e−β(|κ|−|λ|)φ(cb), and (5.4) is equivalent to φ(cb) = eβ(|κ|−|λ|)φ(bc) = e−β(|µ|−|ν|) φ(bc). Thus it suffices to prove (5.4) for φ(bc) 6= 0, because we can then swap b and c and deduce it for φ(cb) 6= 0. Next we observe that it suffices to prove (5.4) when |µ| ≥ |λ|, because taking complex conjugates reduces the other case to this one (see the end of the proof of [17, Proposition 4.1]). So we suppose that φ(bc) 6= 0 and |µ| ≥ |λ|. Then since φ(bc) 6= 0, (4.7) implies that there exists µ′ such that µ = λµ′ . Then the first option in (4.7) shows that φ(bc) = φ(sκ(g·µ′ ) ug|µ′ uh s∗ν ) (5.5)

′

= δκ(g·µ′ ),ν δs(µ),r(g|µ′ ) δs(ν),d(h) e−β(|κ|+|µ |) φ(ug|µ′ uh ).

Thus φ(bc) 6= 0 implies that ν = κ(g · µ′ ), s(µ) = r(g|µ′ ) = d(h) and ′

φ(bc) = e−β(|κ|+|µ |) φ(ug|µ′ uh );

since then ug|µ′ uh 6= 0, we also have d(g|µ′ ) = c(h). Now we remember that ν = κ(g · µ′ ), and use the second option in (4.7) to compute φ(cαiβ (b)) = e−β(|κ|−|λ|) φ(cb)

= e−β(|κ|−|λ|) φ (sµ uh s∗ν )(sκ ug s∗λ )

= e−β(|κ|−|λ|) φ sµ uhg|g−1 ·(g·µ′ ) s∗λ(g−1 ·(g·µ′ )) = e−β(|κ|−|λ|) φ sµ uhg|µ′ s∗λµ′

= e−β(|κ|−|λ|) e−β|µ| δµ,λµ′ δs(µ),c(h) δs(µ),d(g|µ′ ) φ(uh ug|µ′ ). All the delta functions are 1, and we deduce that φ(cαiβ (b)) = e−β(|κ|−|λ|+|µ|)φ(uh ug|µ′ ) = e−β(|κ|−|λ|+|µ|)φ(ug|µ′ uh ),

(5.6) because φ ◦ iC ∗ G) is a trace. Since

|κ| − |λ| + |µ| = |κ| − |λ| + (|λ| + |µ′ |) = |κ| + |µ′ |,

we deduce that (5.5) = (5.6), and we are done.


19

If τ is a trace on a groupoid algebra C ∗ (G), then τ (ig ) = τ (ic(g) ig id(g) ) = τ (ig id(g) ic(g) ),

g ∈ G,

so τ (ig ) 6= 0 implies d(g) = c(g). So in particular we can make the following simple and important observation: Corollary 5.2. Suppose that φ is a KMSβ state of (T (G, E), σ). Then φ(ug ) 6= 0 =⇒ d(g) = c(g). 6. KMS states above the critical temperature Theorem 6.1. Suppose that E is a finite graph with no sources and G is a groupoid which acts self-similarly on E. Let B be the vertex matrix of E, and let σ be the dynamics on T (G, E) such that σt (sµ uh s∗ν ) = eit(|µ|−|ν|) sµ uh s∗ν . Suppose that β > ln ρ(B), where ρ(B) is the spectral radius of B. (1) For each normalised trace τ on C ∗ (G), the series (6.1)

∞ X X

e−βj τ (is(µ) )

j=0 µ∈E j

converges with sum Z(β, τ ), say. (2) For each normalised trace τ on C ∗ (G) there is a KMSβ -state ψβ,τ of (T (G, E), σ) such that ∞ X X ∗ −1 e−βj τ ig|ν . (6.2) ψβ,τ (sκ ug sλ ) = δκ,λ Z(β, τ ) j=|κ|

{ν∈s(κ)E j−|κ| : g·ν=ν}

(3) The map τ 7→ ψβ,τ is an affine homeomorphism of the simplex of tracial states of C ∗ (G) onto the simplex of KMSβ states of (T (G, E), σ). Our strategy for part (2) is based on a construction in [14] (see the proof of Theorem 2.1 in [14]). Given τ , we consider the GNS πτ of C ∗ (G) on Kτ , say. We then L∞ represntation consider the Fock module F (M) = j=0 M ⊗n , which is also a bimodule over C ∗ (G), and the induced representation of C ∗ (G) on F (M) ⊗C ∗ (G) Kτ . We then extend this induced representation to a representation of T (G, E), and construct our KMS states as sums of vector states in this representation. As in previous applications [15, 16, 17], we find it easier to work in a concretely defined realisation of this induced representation. The key observation is that the tensor powers X(E)⊗j of the graph bimodule X(E) used in the construction of M can be realised as functions on the path spaces E j (see [23, Proposition 9.7]). For j ∈ N, we consider the graph bimodule X(E j ) of the graph (E 0 , E j , r, s), which is a Hilbert bimodule over C(E 0 ). Thus we can form the Hilbert space Hj := X(E j )⊗C(E 0 ) Kτ . For a path µ ∈ E j , we also write µ for the characteristic function of the set {µ}. Then every elementPx of C(E j ), which is the underlying space of X(E j ), is the finite linear combination µ∈X j x(µ)µ. Because of the balancing in Hj = X(E j ) ⊗C(E 0 ) Kτ , we have µ ⊗ k = (µ · s(µ)) ⊗ k = µ ⊗ πτ (is(µ) )k for µ ∈ E j , k ∈ Kτ .

20


The Hilbert space of our representation is then H := ∗

L∞

j=0 Hj .

We observe that

H = span{µ ⊗ k : µ ∈ E , k ∈ Kτ and πτ (is(µ) )k = k}.

We observe for future use that these spanning vectors satisfy (6.3) µ ⊗ k | ν ⊗ l = πτ (hν, µi)k | l = δµ,ν (πτ (is(µ) )k | l).

Thus if we choose an orthonormal basis {ev,i } for each πτ (iv )Kτ , then {µ ⊗ es(µ),i } is an orthonormal basis for H. Lemma 6.2. Suppose that E is a finite graph with no sources, and G is a groupoid with unit space E 0 which acts self-similarly on E. Let τ be a tracial state of C ∗ (G), and take πτ and H as above. Then there is a Toeplitz-Cuntz-Krieger family (P, S) on H such that (6.4)

Pv (µ ⊗ k) = δv,r(µ) µ ⊗ k

and

Se (µ ⊗ k) = δs(e),r(µ) eµ ⊗ k;

and there is a unitary representation U : G → U(H) such that for each g ∈ G ( (g · µ) ⊗ πτ (ig|µ )k if r(µ) = d(g) (6.5) Ug (µ ⊗ k) = 0 otherwise. Moreover, the triple (U, P, S) satisfies the relations (1–4) of Proposition 4.4. Proof. Looking at their effects on the orthonormal basis {µ ⊗ es(µ),i } shows that there are projections Pv and a partial isometries Ue satisfying (6.4), and that they form a Toeplitz-Cuntz-Krieger family. Regarding now the vertex v as a unit of the groupoid G, we define Uv := Pv . We now need to define Ug for more general g ∈ G. Each vector h in the algebraic linear span H ′ := span{µ ⊗ k : µ ∈ E ∗ , k ∈ Kτ and πτ (is(µ) )k = k} P has a unique sum decomposition h = µ∈F µ ⊗ kµ for some finite subset F of E ∗ . So we can define Ug as a function on Pd(g) H ′ by X X Ug µ ⊗ kµ := (g · µ) ⊗ πτ (ig|µ )kµ . µ∈F

µ∈F

We aim to prove that Ug is isometric on Pd(g) H ′ , and hence extends to an isometry on Pd(g) H ′ = Pd(g) H. We have

X X 2 X

U )k (g · ν) ⊗ π (i µ ⊗ k = )k (g · µ) ⊗ π (i

g ν τ g|ν µ τ g|µ µ µ∈F

ν∈F

µ∈F

Since hµ, νi = δµ,ν is(µ) , the µ, ν summand on the right-hand side vanishes unless g·µ = g·ν, and hence unless µ = ν. Thus

X 2 X

µ ⊗ kµ = πτ (ig|µ )kµ πτ (ig|µ )kµ .

Ug µ∈F

µ∈F

Since πτ (ig|µ ) is an isomorphism of d(g|µ)E ∗ = s(µ)E ∗ onto c(g|µ)E ∗ , it follows that

X

X

2 2 X

µ ⊗ kµ = (kµ | kµ ) = µ ⊗ kµ .

Ug µ∈F

µ∈F

µ∈F

Thus the map Ug is well-defined and isometric on a dense subset of d(g)H = Ud(g) H, and hence extends to an isometry of d(g)H into Ud(g) H. Since µ 7→ g · µ is onto c(g)E ∗ and


21

πτ (ig|µ ) is onto πτ (ic(g|µ ) )Kτ = πτ (is(g·µ) )Kτ (because πτ is a unitary representation), every elementary tensor in Pc(g) H ′ belongs to the range of Ug . Thus Ug is a unitary isomorphism of Pd(g) H onto Pc(g) H. Now take g, h ∈ G. If d(g) 6= c(h), then the final subspace Pc(h) H of Uh is orthogonal to the initial subspace Pd(g) H of Ug , and hence Ug Uh = 0. So we assume that d(g) = c(h). Then the range projection of Uh is the initial projection of Ug , and thus Ug Uh is a partial isometry whose initial and final projections are the same as those of Ugh . To prove that Ug Uh = Ugh , we compute on an elementary tensor µ ⊗ k with r(µ) = d(g): Ug Uh (µ ⊗ k) = Ug (h · µ ⊗ π(ih|µ )k) = g · (h · µ) ⊗ π(ig|h·µ )π(ih|µ )k = gh · µ ⊗ π(igh|µ ) = Ugh (µ ⊗ k).

Thus U is a unitary representation of the groupoid G on H. Next we take g ∈ G and e ∈ E 1 , and verify relation (3) in Proposition 4.4. If d(g) 6= r(e), then Se H = 0 = Pr(e) H is orthogonal to the initial space Pd(g) H of Ug , and hence Ug Se = 0. So suppose that d(g) = r(e), and take an elementary tensor µ ⊗ k ∈ H with s(e) = r(µ). Then we compute Ug Se (µ ⊗ k) = Ug (eµ ⊗ k) = (g · eµ) ⊗ π(ig|eµ )k

= (g · e)(g|e · µ) ⊗ π(ig|eµ )k = Sg·e (g|e · µ ⊗ π(ig|e |µ )k)

= Sg·e Ug|e (µ ⊗ k).

Finally, we check that Ug Pv = Pg·v Ug . For µ ⊗ k ∈ H, we have Ug Pv (µ ⊗ k) = Ug (δv,r(µ) µ ⊗ k) = δv,r(µ) g · µ ⊗ π(ig|µ )k

= δg·v,r(g·µ) g · µ ⊗ π(ig|µ )k = Pg·v (g · µ ⊗ π(ig|µ )k) = Pg·v Ug (µ ⊗ k).

Thus (U, P, S) satisfies all the relations in Proposition 4.4, as required.

Proof of Theorem 6.1(1). Suppose that τ is a normalised trace on C ∗ (G), and consider the corresponding GNS representation (Kτ , πτ , ξτ ) of C ∗ (G). Construct the Hilbert space H as above. Let πU,S,P : T (G, E) → B(H) be the representation arising from the family (U, P, S) of Lemma 6.2 by the universal from Proposition 4.4. P P property −βj Suppose that β > ln ρ(B). Then ∞ converges for each v ∈ E 0 by [10, j=0 µ∈E j v e Theorem 3.1(a)]. Since 0 ≤ τ (is(µ) ) ≤ 1 for every µ ∈ E ∗ , it follows that (6.6)

∞ X X

−βj

e

τ (is(µ) ) =

∞ X XX

e−βj τ (is(µ) )

v∈E 0 j=0 µ∈E j v

j=0 µ∈E j

converges.

Proof of Theorem 6.1(2). We begin by defining ψβ,τ spatially using the representation on H. As in the previous paragraph, for every a ∈ T (G, E), the series (6.7)

ψβ,τ (a) := Z(β, τ )

−1

∞ X j=0

e−βj

X

µ∈E j

πU,P,S (a)(µ ⊗ ξτ ) | µ ⊗ ξτ

converges absolutely with sum ψβ,τ (a) ∈ C. By standard properties of convergent series, the assignment ψβ,τ : a 7→ ψβ,τ (a) is a positive linear functional on T (G, E). In fact, we

22


have ψβ,τ (1) = Z(β, τ )

−1

∞ X

e−βj

j=0

= Z(β, τ )−1

∞ X j=0

= 1,

X

µ∈E j

e−βj

X

(µ ⊗ ξτ | µ ⊗ ξτ ) τ (is(µ) )

µ∈E j

and hence ψβ,τ is a state. To prove that ψβ,τ satisfies (6.2), we fix a spanning element sκ ug s∗λ . Then ∞ X X (6.8) ψβ,τ (sκ ug s∗λ ) = Z(β, τ )−1 e−βj Sκ Ug Sλ∗ (µ ⊗ ξτ ) µ ⊗ ξτ . j=0

µ∈E j

Suppose there exists µ such that

Sκ Ug Sλ∗ (µ ⊗ ξτ ) | µ ⊗ ξτ = Ug Sλ∗ (µ ⊗ ξτ ) | Sκ∗ (µ ⊗ ξτ ) 6= 0.

Since Sκ∗ (µ ⊗ ξτ ) 6= 0, there exists µ′ such that µ = κµ′ , and similarly there exists µ′′ such that µ = λµ′′ . Then 0 6= Ug Sλ∗ (µ ⊗ ξτ ) | Sκ∗ (µ ⊗ ξτ ) = Ug (µ′′ ⊗ ξτ ) | µ′ ⊗ ξτ (6.9) = (g · µ′′ ) ⊗ πτ (ig|µ′′ )ξτ | µ′ ⊗ ξτ ,

which implies that g · µ′′ = µ′ and |µ′′ | = |g · µ′′ | = |µ′|. Now we have κµ′ = µ = λµ′′ . From all this we deduce: first, that κ = λ, which gives the δκ,λ in (6.2); second, that only the terms with µ = κµ′ contribute non-zero terms to the sum on the right of (6.8); third, that only the terms with g · µ′ = µ′ survive. For these terms, we have Sκ Ug Sλ∗ (µ ⊗ ξτ ) | µ ⊗ ξτ = g · µ′ ⊗ πτ (ig|µ′ )ξτ | µ′ ⊗ ξτ = πτ (is(µ) ig|µ′ )ξτ ) | µ′ ⊗ ξτ = πτ (ig|µ′ )ξτ ) | ξτ .

Now summing over µ′ ∈ s(κ)E ∗ satisfying g · µ′ = µ′ , and writing ν for µ′ , gives (6.2). To show that ψβ,τ is a KMSβ state, we verify the conditions in Proposition 5.1. To see that ψβ,τ ◦ iC ∗ (G) is a trace on C ∗ (G), consider two generators ig and ih in C ∗ (G). Then (6.2) gives ψβ,τ ◦ iC ∗ (G) (ig ih ) = ψβ,τ (ug uh ) = δd(g),c(h) ψβ,τ (ugh ) ∞ X −1 (6.10) = δd(g),c(h) δc(g),d(h) Z(β, τ ) e−βj

X

{ν∈d(h)E j :gh·ν=ν}

τ i(gh)|ν .

ψβ,τ ◦ iC ∗ (G) (ih ig ) = ψβ,τ (uh ug ) = δd(h),c(g) ψβ,τ (uhg ) ∞ X −1 = δd(h),c(g) δc(h),d(g) Z(β, τ ) e−βj (6.11)

X

τ i(hg)|µ .

j=0

On the other hand, we have

j=0

{µ∈d(g)E j :hg·µ=µ}

We need to prove that (6.10) = (6.11). Both sides vanish unless d(h) = c(g) and c(h) = d(g), so we suppose that both are true. We claim that the map θ : µ → 7 g·µ


23

is a bijection of the index set {µ ∈ d(g)E j : (hg) · µ = µ} in the sum (6.11) onto the index set {ν ∈ d(h)E j : (gh) · ν = ν} in (6.10). Suppose µ ∈ d(g)E j satisfies hg(µ) = µ. Then r(µ) = d(g) gives r(g · µ) = c(g) = d(h), so g · µ ∈ d(h)E j . Now (gh)·(g ·µ) = g ·(hg ·µ) = g ·µ, and hence g ·µ belongs to the set {ν ∈ d(h)E j : gh·ν = ν}. The map ν 7→ h · ν is an inverse for θ, and the claim follows. Now we show that the bijection θ matches up the summands as well as their labels. Suppose that ν ∈ d(h)E j satisfies gh · ν = ν. Then by Proposition 3.6 τ i(gh)|ν = τ ig|h·ν ih|ν since τ is a trace = τ ih|ν ig|h·ν because gh · ν = ν = τ ih|gh·ν ig|h·ν = τ i(hg)|h·ν .

Thus the sums in (6.10) and (6.11) coincide, and ψβ,τ ◦ iC ∗ (G) is a trace on C ∗ (G). We now prove that ψβ,τ satisfies (5.3). Let sκ ug s∗λ ∈ T (G, E) and assume that s(κ) = g · s(λ), for otherwise the product vanishes. On one hand, (6.2) gives ψβ,τ (sκ ug s∗λ ) = δκ,λ Z(β, τ )−1

∞ X

e−βj

j=|κ|

= δκ,λ Z(β, τ )−1

∞ X

X

τ ig|ν

{ν∈s(κ)E j−|κ| :g·ν=ν}

e−β|κ| e−βk

k=0

X

{ν∈s(κ)E k :g·ν=ν}

τ ig|ν .

On the other hand, since ψβ,τ ◦ iC ∗ (G) is a trace, the right hand side of (5.3) gives δκ,λ e−β|κ| ψβ,τ (ps(κ) ug ) = δκ,λ e−β|κ| ψβ,τ (ps(κ) ug ps(κ) ) ∞ X −β|κ| −1 e−βk = δκ,λ e Z(β, τ ) k=0

= δκ,λ Z(β, τ )

−1

∞ X

X


e−β|κ| e−βk

k=0

X


τ is(ν) ig|ν

τ is(ν) ig|ν .

Since g · ν = ν implies that is(ν) ig|ν = ig|ν , we deduce that ψβ,τ satisfies (5.3). Now Proposition 5.1 implies that ψβ,τ is a KMSβ -state. This finishes the proof of Theorem 6.1(2). The key step in our proof of part (3) of Theorem 6.1 is proving that every KMS state has the form ψβ,τ for exactly one trace τ on C ∗ (G). The proof of uniqueness follows previous ones (for example, in [15, §10] and [17, §6]) in which a KMSβ state is reconstructed from its conditioning by a projection in T (G, E). Lemma 6.3. Suppose that β > ln ρ(B) and φ is a KMSβ state of T (G, E). Define 0 0 mφ ∈ [0, ∞)E by mφv := φ(pv ) in [0, ∞)E , and a projection P in T (G, E) by X X X se s∗e . pv − se s∗e = P := 1 − e∈E 1

v∈E 0

e∈vE 1

24


Then (6.12)

φ(P ) =

X

v∈E 0

(1 − e−β B)mφ

v

> 0,

and φP : a 7→ φ(P )−1 φ(P aP ) is a state on T (G, E) such that φP ◦ iC ∗ (G) is a trace on C ∗ (G). Proof. A computation like that of [10, Equation (2.5)] yields X X ∗ φ(se se ) φ(pv ) − φ(P ) = e∈vE 1

v∈E 0

=

X

v∈E 0

=

X

v∈E 0

=

X

φ(pv ) − φ(pv ) −

X

e−β φ(s∗e se )

e∈vE 1

X

w∈E 0

|vE 1 w|e−β φ(pw )

(1 − e−β B)mφ v .

v∈E 0

P Since 1 = φ(1) = v φ(pv ), the vector mφ is nonzero, and the matrix 1−e−β B is invertible because eβ > ρ(B). Thus φ(P ) > 0, and φP is a state. With a view to proving that φP is a trace on the copy of C ∗ (G), we first claim that if d(g) = r(e), then ug se s∗e = sg·e s∗g·e ug . Theorem 4.4 imples that ug se = sg·e ug|e . Now we use the properties of Proposition 3.6 to compute )∗ = (se ug−1 |g·e )∗ ug|e s∗e = (se u∗g|e )∗ = (se u|g|−1 e

= (sg−1 ·(g·e) ug−1 |g·e )∗ = (sg−1 ·(g·e) ug−1 |g·e )∗ = (ug−1 sg·e )∗ = s∗g·e ug .

Thus ug se s∗e = sg·e ug|e s∗e = sg·e s∗g·e ug , as claimed. Now for g ∈ G we have X X ∗ ug se s∗e se se = u g − ug P = ug 1 − e∈g(g)E 1

e∈E 1

= ug −

X

sg·e s∗g·e ug .

e∈d(g)E 1

Since e 7→ g · e is a bijection of d(g)E 1 onto c(g)E 1 , we have X X sf s∗f ug = P ug . sf s∗f ug = 1 − ug P = 1 − f ∈c(g)E 1

f ∈E 1

Suppose that g, h ∈ G satisfy d(g) = c(h). Then ug and ug P are fixed under the action σ, so the KMS condition implies that φ(P ug uh P ) = φ(ug P 2 uh ) = φ(P uh σiβ (ug P )) = φ(P uh ug P ). Thus a 7→ φ(P aP ) is a trace on C ∗ (G), and so is the multiple φP . The next lemma is the main technical ingredient in our reconstruction formula.


25

Lemma 6.4. Suppose that β > ln ρ(B) and φ is a KMSβ state on (T (G, E), σ). Then for a ∈ T (G, E), we have (6.13)

φ(a) = φ(P )

∞ X X

e−βj φP (s∗µ asµ )

j=0 µ∈E j

=

X

v∈E 0

−β

(1 − e

φ

B)m

∞ X X

e−βj φP (s∗µ asµ ).

v

j=0 µ∈E j

Proof. A computation similar to the one in the proof of [17, Lemma 6.3] shows that for each n ∈ N n X X pn := sµ P s∗µ j=0 µ∈E j

is a projection in T (G, E). We next show that φ(pn ) → 1 as n → ∞. To do this, we compute: φ(pn ) = =

n X X

j=0 µ∈E j n X X

j=0 µ∈E j

=

n X

X

j=0 µ∈E j

φ(sµ P s∗µ )

=

n X X

e−βj φ(P s∗µ sµ )

j=0 µ∈E j

e−βj φ(ps(µ) ) −

e−βj mφs(µ) −

X

φ(se s∗e )

r(e)=s(µ)

X

r(e)=s(µ)

e−β mφs(e) .

Now we rewrite this in terms of the vertex matrix B, and continue the computation, using some tricks from [10, §3]: n X X X φ −β φ −βj e B(s(µ), w)mw ms(µ) − e φ(pn ) = = = = =

j=0 µ∈E j n X X j=0 µ∈E j n X X

w∈E 0

e−βj (1 − e−β B)mφ

s(µ)

X

e−βj B j (v, w) (1 − e−β B)mφ

e

B j (1 − e−β B)mφ

j=0 v∈E 0 w∈E 0 n X X −βj j=0 v∈E 0 n X X

v∈E 0

j=0

−βj

e

j

−β

B (1 − e

φ

B)m

w

v

v

.

P Since β > ln ρ(B), we have eβ > ρ(B), and the series j e−βj B j converges with sum (1 − e−β B)−1 . Thus as n → ∞ X X mφv = 1. (1 − eβ B)−1 (1 − e−β B)mφ v = φ(pn ) → v∈E 0

v∈E 0

26


Since φ(pn ) → 1, we have φ(pn apn ) → φ(a) as n → ∞ (see, for example, [16, Lemma 7.3]). For µ, ν ∈ E j , the elements sµ P s∗µ and sν P s∗ν belong to the fixed-point algebra for σ, and hence since φ is a KMS state, we have φ (sµ P s∗µ )a(sν P s∗ν ) = 0 for a ∈ T (G, E) and µ 6= ν. We now use the KMS condition again to compute ∞ X X φ(a) = φ(sµ P s∗µ asµ P s∗µ ) = = =

j=0 µ∈E j ∞ X X



e−βj φ(P s∗µ asµ P s∗µ sµ ) e−βj φ(P s∗µ asµ s∗µ sµ P ) e−βj φ(P )φP (s∗µ asµ ).

j=0 µ∈E j

The second formula follows from this and (6.12).

Proof of Theorem 6.1(3). The proof follows that of [17, Theorem 6.1]. An application of the monotone convergence theorem shows that τ 7→ ψβ,τ is affine and weak∗ continuous. Since the set of tracial states and the set of KMSβ states are both weak∗ compact, it suffices to show that the map τ 7→ ψβ,τ is one-to-one and onto. The equations (6.4) in Lemma 6.2 imply that the representation π = πU,P,S of T (G, E) satisfies π(P )(µ ⊗ ξτ ) = 0 unless µ = v ∈ E 0 , in which case π(P )v ⊗ ξτ = v ⊗ ξτ . We now take b ∈ C ∗ (G), and set a = P u(b)P in (6.7). Then all the terms with j > 0 vanish, yielding X (6.14) ψβ,τ (P u(b)P ) = Z(β, τ )−1 τ (pv bpv ) = Z(β, τ )−1 τ (b). v

−1

Taking b = 1 shows that ψβ,τ (P ) = Z(β, τ ) , and then dividing (6.14) through by ψβ,τ (P ) gives (6.15)

(ψβ,τ )P (u(b)) = ψβ,τ (P )−1 ψβ,τ (P u(b)P ) = τ (b).

Thus τ = (ψβ,τ )P ◦ u, and the map τ 7→ ψβ,τ is one-to-one. It remains to show that the map τ 7→ ψβ,τ is onto. Suppose that φ is a KMSβ -state on T (G, E). By Lemma 6.3, τ := φP ◦ u is a tracial state on C ∗ (G), and by the above argument, φP ◦ u = τ = (ψβ,τ )P u. Thus Lemma 6.4 implies that φ(ug ) = ψβ,τ (ug ) for every g ∈ G, and in Proposition 5.1(2) we have φ = ψβ,τ . 7. Traces on groupoid algebras Theorem 6.1 tells us that the KMS states at large inverse temperatures are parametrised by traces on the groupoid algebra C ∗ (G). In general finding these traces is a hard problem, even for group algebras — indeed, one unexpected outcome of our previous analysis of KMS states in was the discovery of new tracial states on group algebras [17, Corollary 7.5]. b The exception is when G is an abelian group, in which case C ∗ (G) is the algebra C(G)


27

b and the tracial states are given by of continuous functions on the compact dual group G, b Fortunately, for the groupoids of interest to us, we can often probability measures on G. reduce to this case. The existence of the isomorphism in the following result is a special case of Theorem 3.1 of [18], but it will be helpful to have a concrete description of the isomorphism. As a local convention to avoid complicated subscripts, if u is a unitary representation of a group or groupoid, we write u(g) rather than ug . Lemma 7.1. Suppose that G is a groupoid with finite unit space G0 and that G acts transitively on G0 . Fix x ∈ G0 , let Gx := xGx be the isotropy group, and write iGx for the canonical unitary representation of Gx in its group algebra C ∗ (Gx ). For each y ∈ G0 choose ky ∈ G such that d(ky ) = x, c(ky ) = y and kx = x. Then there is an isomorphism ψ of MG0 (C ∗ (Gx )) onto C ∗ (G) such that X i(ky )πi (ayz )i(kz )−1 . (7.1) ψ (ayz ) = y,z∈G0

The inverse ψ −1 satisfies (7.2)

−1

ψ (i(g))yz

( −1 u kc(g) gkd(g) if y = c(g) and z = d(g) = 0 otherwise.

Proof. We verify that {i(ky kz−1 ) : y, z ∈ G0 } is a set of non-zero matrix units in C ∗ (G), and hence there is an injective homomorphism π : MG0 (C) → C ∗ (G) which takes the usual matrix units eyz to i(ky kz−1 ). The restriction i|Gx is a unitary representation (in the usual group-theoretic sense) of Gx in the corner i(x)C ∗ (G)i(x), and hence there is a unital homomorphism ρ : C ∗ (Gx ) → i(x)C ∗ (G)i(x) such that ρ◦u = i|Gx . To see that ρ is injective, we show that every unitary representation U : Gx → U(H) is the compression to Gx of a unitary representation W of the groupoid G. Indeed, we take Hx = H, and for y ∈ G0 let Hy be copies of H, with Vy : H → Hy the identity maps. Then for g ∈ G, −1 ∗ W (g) := Vc(g) U(kc(g) gkd(g) )Vd(g) : Hd(g) → Hc(g)

is a unitary isomorphism. For g, h ∈ G with d(g) = c(h) we have Vd(g) = Vc(h) and kd(g) = kc(h) . Thus −1 ∗ −1 ∗ W (g)W (h) = Vc(g) U(kc(g) gkd(g) )Vd(g) Vc(h) U(kc(h) hkd(h) )Vd(h) −1 = Vc(g) U(kc(g) ghkd(h) )Vd(h) .

Now c(gh) = c(g) and d(gh) = d(h). Thus −1 W (g)W (h) = Vc(gh) U(kc(gh) ghkd(gh) )Vd(gh) = W (gh), L and W is a unitary representation of the groupoid G on y∈E 0 Hy , with W (x) the projection on the Hilbert space H of U. Since kx = x and Vx is the identity, we have W (x)W (g)W (x)∗ = U(g) for g ∈ Gx . Thus the homomorphism ρ is isometric for the enveloping P C ∗ -norms on Cc (Gx ) and Cc (G), and in particular is injective. It follows that θ : u(g) 7→ y∈G0 i(ky gky−1 ) is also injective from C ∗ (Gx ) to C ∗ (G).

28


Now we show that θ and π have commuting ranges. Indeed, for g ∈ Gx and a matrix unit ezw ∈ Mn (C), X i(ky gky−1)i(kz gkw−1 ) θ(iGx (g))π(ezw ) = y∈G0 i(kz gkw−1 );

vanishes unless y = z, and then equals a similar computation on the other side shows that only the y = w summand in π(ezw )θ(iGx (g)) survives, and is again equal to i(kz gkw−1). Thus θ and π have commuting ranges, and there is an injection θ ⊗max π of the maximal tensor product C ∗ (Gx ) ⊗max MG0 (C) into C ∗ (G) such that

(7.3)

θ ⊗max π(iGx (g) ⊗ eyz ) = θ(iGx (g))π(eyz ) = i(ky gkz−1)

(see [24, Theorem B.27]). Since each

−1 −1 −1 i(h) = i kc(h) (kc(h) hkd(h) )kd(h) ) = θ ⊗max π(u(kc(h) hkd(h) ) ⊗ ec(h)d(h) )

belongs to the range of θ ⊗max π, it is an isomorphism of C ∗ (Gx ) ⊗max MG0 (C) onto C ∗ (G). When P we view C ∗ (Gx ) ⊗max MG0 (C) as MG0 (C ∗ (Gx )), we identify the matrix (ayz ) with the sum y,z ayz ⊗ i(eyz ) (see [24, Proposition B.18]). Hence the formulas (7.1) and (7.2) follow from (7.3).

Proposition 7.2. Suppose that G is a groupoid with finite unit space G0 and that G acts transitively on G0 . Fix x ∈ G0 , let Gx be the isotropy group at x, and choose elements ky ∈ G such that d(ky ) = x, c(ky ) = y and kx = x. Then for every trace τ on C ∗ (Gx ), there is a trace τ ′ on C ∗ (G) such that ( −1 τ u(kd(g) gkd(g) ) if d(g) = c(g) ′ (7.4) τ (i(g)) = 0 otherwise. The trace τ ′ does not depend on the choice of orbit representatives {ky }.

Proof. There is a trace σ on MG0 (C ∗ (Gx )) such that X τ (ayy ). σ (ayz ) = y∈G0

Pulling this over to C ∗ (G) via the isomorphism of Lemma 7.1 gives a trace τ ′ on C ∗ (G) satisfying X τ ψ −1 (i(g))yy τ ′ (i(g)) = σ ψ −1 (i(g)) = y∈G0

( −1 τ (u(kd(g) gkd(g) )) if d(g) = c(g) = 0 otherwise,

as required. 0 If {hy : y ∈ G0 } is another set of orbit representatives, then h−1 y ky ∈ Gx for all y ∈ G . Thus if d(g) = c(g), say d(g) = y = c(g), we have −1 −1 −1 τ (u(h−1 y ghy )) = τ u(hy ky ky gky ky hy ) −1 −1 −1 = τ u(h−1 y ky )u(ky gky )u(hy ky ) −1 −1 because τ is a trace = τ u(ky−1 gky )u(h−1 y ky ) u(hy ky ) = τ u(ky−1 gky )).


29

Corollary 7.3. In the situation of Proposition 7.2, there are traces τe′ and τ1′ on C ∗ (G) such that ( 1 if g ∈ G0 τe′ (i(g)) = 0 otherwise, and ( 1 if d(g) = c(g) τ1′ (i(g)) = 0 otherwise. Proof. The isotropy group algebra C ∗ (Gx ) has (at least) two traces τe and τ1 (see the discussion on page 6648 of [17], for example), and we can apply Proposition 7.2 to both −1 of them. The elements kc(g) gkd(g) of G all belong to Gx , and hence the formulas for τe′ and τ1′ follow from (7.4) and the corresponding properties of τe and τ1 : τe (i(g)) = 0 for g 6= idx = eGx , and τ1 (g) = 1 for all g ∈ Gx . Remark 7.4. Both traces τe′ and τ1′ can be spatially implemented. For τe′ , take the regular unitary representation λ of G on ℓ2 (G), defined in terms of the usual orthonormal basis {ξg : g ∈ G} by ( ξgh if d(g) = c(h) λg ξ h = 0 if d(g) 6= c(h). Then

τe′ (a) =

X

x∈G0

τ1′ ,

(πλ (a)ξx | ξx ) for a ∈ C ∗ (G).

For we take the unitary representation U of G on ℓ2 (G0 ) from Example 4.2. Then in terms of the usual basis {hx : x ∈ G0 }, we have X (πU (a)hx | hx ) for a ∈ C ∗ (G). τ1′ (a) = x∈G0

Now suppose that G is a groupoid with finite unit space G0 . We define a relation ∼ on G0 by x ∼ y ⇐⇒ xGy 6= ∅. Then ∼ is an equivalence relation, and we write G0 /∼ for the set of equivalence classes. For each component C ∈ G0 /∼, we let GC := {g ∈ G : d(g) ∈ C and c(g) ∈ C}

be the reduction of G to C.

Lemma 7.5. Let i : G → C ∗ (G) be the canonical unitary representation of G. Then i|GC is a unitary representation homomorphism from L of GC ; let πC be the corresponding P ∗ ∗ C (GC ) into C (G). Then C∈G0 /∼ πC : (aC ) 7→ C πC (aC ) is an isomorphism of L ∗ ∗ C∈G0 /∼ C (GC ) onto C (G).

Proof. Since GC is closed under all the operations of G, including the domain and codomain maps1, the restriction i|GC is a unitary representation of GC . Every unitary representation U of GC extends to a unitary representation of G, simply by taking Ug = 0 for g ∈ / GC , and hence factors through the homomorphism πC ; thus πC is injective. For D ∈ G0 /∼ and D 6= C, we have iD (h)iC (g) = (iD (h)i(d(h)))(i(c(g))iD (g)) = 0, and hence the ranges of πD and πC satisfy πD (C ∗ (GD ))πC (C ∗ (GC )) = 0. Since every g ∈ G belongs to some 1Normally

one would say that GC is a subobject of G. But we are using “subgroupoid” to mean a subobject with the same unit space.

30


GC (the one forLwhich d(g) ∈ C and c(G) ∈ [23, Proposition A.6], for LC),∗ it follows from example, that πC is an isomorphism of C (GC ) onto C ∗ (G).

Corollary 7.6. Suppose that G is a groupoid with finite unit space G0 , and C ∈ G0 /∼. Fix x ∈ C and choose representatives ky for each xCy with y ∈ C. Then for every trace τC on C ∗ (Gx ), there is a trace τC′ on C ∗ (G) such that ( −1 τC (u(kc(g) gkd(g) )) if d(g) = c(g) ∈ C (7.5) τC′ (i(g)) = 0 otherwise. Every trace on C ∗ (G) is a sum of traces of the form τC′ . Proof. The groupoid GC is transitive, so Proposition 7.2 gives us a trace τC′ on C ∗ (GC ). L ∗ We can extend this trace to D C (GD ) by taking it to be zero on the other summands, and then use the isomorphism of Lemma 7.5 to pull it over to a trace τ ′ on C ∗ (G). Since the isomorphism on the summand C ∗ (GC ) comes from i|GC , the formula (7.5) follows from ∗ (7.4). For the last remark, note that P if τ is a′ trace on C (G), then τ ◦ πC is a trace on ∗ C (GC ), and we can recover τ as C (τ ◦ πC ) .

At this point, we discuss an important class of examples that motivated the previous work of Exel and Pardo [8]. The Cuntz-Pimsner algebras of the following self-similar groupoids are a family of algebras constructed by Katsura [13] to provide models of Kirchberg algebras. These examples also fit the theory of [8], and in particular have the property s(g · e) = s(e) = g · s(e).

Example 7.7. Suppose that N ∈ N, and consider matrices A = (aij ), B = (bij ) in MN (N) such that A has no nonzero rows and aij = 0 =⇒ bij = 0. Let E be the graph with E 0 = {1, 2, · · · , N}, E 1 = {ei,j,m : 0 ≤ m < aij }, r(ei,j,m) = i, s(ei,j,m) = j. We define an automaton A over E as follows. Let2 A := {ai : 1 ≤ i ≤ N} with s(ai ) = i and r(ai ) = i. We then define, for µ ∈ jE ∗ , (7.6)

ai · ei,j,mµ = ei,j,n (alj · µ) where bij + m = laij + n and 0 ≤ n < ai .

We consider the faithful self-similar groupoid action (GA , E) of Theorem 3.9. Each generator fai of GA acts trivially on the vertex set E 0 , and hence so does every element of GA . Thus the connected components C ∈ E 0 /∼ are the singletons Ci = {i}. The reductions (GA ){i} each contain one generator fai , and hence they act on different subtrees of TE . Thus every element of GA is a power of one generator fai , and the reductions (GA ){i} are cyclic groups generated by fai . For each i, there is either an integer ni such that (GA ){i} = {i, fai , fa2i , · · · , fanii−1 }, or (GA ){i} = {faki : k ∈ Z}. In the first case, the normalised traces on C ∗ ((GA ){i} ) = Cn form a simplex ∆i of dimension ni − 1, in the second case they form a copy ∆i of the probability measures on the circle T. Corollary 7.6 ′ then says that for each i and each µi ∈ ∆i , there is a distinct normalised trace τi,µ on i ∗ C (GA ). Theorem 6.1 says that each gives a distinct KMSβ state on T (GA , E). 2We

have used the calligraphic A to distinguish the automaton from the matrix A in the definition of the Katsura action, thereby avoiding a clash with the established notation in [13] and [8].


31

Example 7.8. We return to the self-similar action described in Example 3.10, and the associated subgroupoid G = GA of PIso(E ∗ ). With E listed as {v, w}, the vertex matrix is ! 1 1 B= , 2 0 which has spectrum σ(B) = {−1, 2} and ρ(B) = 2. So Theorem 6.1 describes the KMSβ states for β > ln 2 in terms of the normalised traces on C ∗ (G). We aim to apply our results to find these. Since a · v = w, the equivalence ∼ on E 0 = {v, w} has a single orbit. Thus Proposition 7.2 gives us a bijection between traces τ on the group algebra C ∗ (Gv ) and traces τ ′ on C ∗ (G). To get a specific formula for τ ′ , we take kv = idv , as instructed for the base point v, and kw = fa . Then we have τ ′ (i(idv )) = τ (u(id−1 v idv ) = τ (u(idv )), and τ ′ (i(idw )) = τ (u(fa−1 idw fa ) = τ (u(idv )). So each trace on C ∗ (Gv ) such that τ (idv ) = 12 gives a normalised trace τ ′ on C ∗ (G). To describe the KMS states, we need to understand C ∗ (Gv ). Recall that G is the subgroupoid of PIso(E ∗ ) generated by two partial isomorphisms fa , fb with d(fa ) = v = c(fb ) and c(fa ) = w = d(fb ). Every g ∈ Gv can be writen g = g1 g2 · · · gn with gi ∈ S := {fa , fb , fa−1 , fb−1 }, gi+1 6= gi−1 and c(g1 ) = v = d(gn ). Since each element of S switches the vertices v and w, n has to be even, say n = 2k. Then c(g) = c(g1 ) = v forces g1 ∈ {fb , fa−1 }. First suppose that g1 = fb . Then g2 has to be in {fa , fb−1 }, and since g2 6= g1−1 , we must have g2 = fa . Continuing this way shows that g = (fb fa )k . On the other hand, if g1 = fa−1 , we have g2 = fb−1 , and continuing gives g = (fa−1 fb−1 )k = (fb fa )k . Thus Gv = {(fb fa )k : k ∈ Z}. Then k 7→ (fb fa )k is either an isomorphism of Z onto Gv or a quotient map, in which case there exists k ∈ N such that (fb fa )k = eGv = idv . We will show that there is no such k, and hence Gv is a copy of Z. So we suppose that k ∈ N, and have to prove that (fb fa )k 6= idv . First we use the defining relations (fb fa )(1) = fb (fa (1)) = fb (a · 1) = fb (4) = b · 4 = 2, and similarly (fb fa )(2) = 1. Thus (fb fa )k is not the identity for k odd. So we suppose that k = 2n is even. Next we do some calculations with the restriction map: (fb fa )|1 = fb|a·1 fa|1 = fb|4 idv = fb|4 = fa , and (fb fa )|2 = fb|3 fa|2 = idv fb = fb , which imply (fb fa )2 |1 = (fb fa )|(fb fa )(1) (fb fa )|1 = (fb fa )|2 fa = fb fa . Continuing this way, we find that (fb fa )2n |1 = (fb fa )n . Now by looking at the action of (fb fa )2n on sufficiently long words of the form µ = 111 · · · 1 we can reduce 2n repeatedly

32


by factors of 2, and arrive at (fb fa )2n |µ = (fb fa )m with m odd, so that (fb fa )2n · µ1 = µ(fb fa )m (1) = µ2. For example, (fb fa )4 (111) = (fb fa )4 (1)(fb fa )4 |1 (11) = (fb fa )4 (fb fa )2 (11) = 1(fb fa )2 (1)(fb fa )2 |1 (1) = 11(fb fa )(1) = 112.

So whatever k is, (fb fa )k acts nontrivially on 1E ∗ . Thus Gv is isomorphic to Z, and the traces of norm 21 on C ∗ (Gv ) are in one-to-one correspondence (via multiplication by 2) with the set of probability measures on the unit circle T. Thus so is the simplex of KMSβ states on (T (G, E), σ) for every β > ln 2. There are two families of self-similar groupoids for which complete classifications of KMS states have previously been found. The first is when |E 0 | = 1, in which case our dynamical system is the one studied in [17]. The second concerns the groupoid G = C(E 0 ), in which every morphism is the identity morphism at some vertex. For this groupoid, for v ∈ E 0 and for g = idv , Proposition 6.4(1) says that ug = pv , and hence T (G, E) is universal for Toeplitz-Cuntz-Kreiger E-families. Thus T (G, E) = T C ∗ (E), and the dynamical system is the system (T C ∗ (E), α) studied in [10]. As a reality check we reconcile our new results with those of [10]. We consider a finite directed graph E with vertex matrix B, and consider a state φǫ 0 associated to a vector ǫ ∈ Σβ ⊂ [0, ∞)E , as in [10, Theorem 3.1]. Equation (3.1) in [10] says that (7.7) φǫ (sκ s∗λ ) = δκ,λ e−β|κ| ms(κ) = δκ,λe−β|κ| (1 − e−β|κ| B)ǫ s(κ) . We look for a normalised trace τ on C ∗ (G) = C(E 0 ) such that φǫ = ψβ,τ . The Banachspace dual of C(E 0 ) is ℓ1 (E 0 ), and the states on C(E 0 ) are given by vectors τ = ( τ (pv ) ) 0 in [0, ∞)E with kτ k1 = 1. Equation (6.2) says that ∞ X X −1 ∗ τ ig|µ e−βj ψβ,τ (sκ sλ ) = δκ,λ Z(β, τ ) j=|µ|

−1 −β|κ|

= δκ,λ Z(β, τ ) e

∞ X

{µ∈s(κ)E j−|κ| : g·µ=µ}

X

−βk

e

k=0

τ ig|µ

{µ∈s(κ)E k : g·µ=µ}

.

So we suppose κ = λ. Then g is the vertex s(κ), g · µ = µ for all µ ∈ s(κ)E k , and s(κ)|µ = s(µ). Thus ∞ X X ∗ −1 −β|κ| ψβ,τ (sκ sλ ) = δκ,λ Z(β, τ ) e e−βk τ (ps(µ) ) k=0

= δκ,λ Z(β, τ )−1 e−β|κ|

∞ X

µ∈s(κ)E k

e−βk

v∈E 0

k=0

−1 −β|κ|

= δκ,λ Z(β, τ ) e

∞ X k=0

X

−βk

e

X

v∈E 0

|s(κ)E k |τ (pv )

B (s(κ), v)τ (pv ) . k

Viewing τ as the vector with entries τ (pv ), we recognise the inner sum as the value of the P∞ k matrix product B τ at the vertex s(κ). Because β > ln ρ(B), the series k=0 e−βk B k τ


converges with sum (1 − e−β B)−1 τ , and hence (7.8)

ψβ,τ (sκ s∗λ ) = δκ,λ Z(β, τ )−1 e−β|κ| (1 − e−β B)−1 τ

s(κ)

33

.

When we compare (7.7) and (7.8), we see that, apart from the factor Z(β, τ )−1, they look similar. However, if they are to be exactly the same, then ǫ has to be a scalar multiple of τ . The key observation is that the only positive scalar multiple of ǫ which gives a state is kǫk−1 1 ǫ. More formally, we have: Proposition 7.9. Suppose that E is a finite graph with vertex B and that β > P matrix β E0 β −β|µ| ln ρ(B). Let y be the vector in [1, ∞) with entries yv = µ∈E 0 v e . Suppose that −1 E0 β ǫ ∈ [0, ∞) satisfies y · ǫ = 1. Then τ := kǫk1 ǫ is a normalised trace on C(E 0 ) satisfying Z(β, τ ) = kǫk−1 1 , and φǫ = ψβ,τ . Conversely, if τ is a normalised trace on C(E 0 ), then ǫ := Z(β, τ )−1 τ satisfies y β · ǫ = 1, and φǫ = ψβ,τ . The key to the proof is the following computation. 0

Lemma 7.10. Resume the notation of Proposition 7.9. Then for every ǫ ∈ [0, ∞)E , we have Z(β, ǫ) = y β · ǫ. Proof. We compute Z(β, ǫ) = = = =

∞ X X

k=0 µ∈E k ∞ X X

X

e−βk B k (w, v)ǫv

e

(B k ǫ)w

k=0 w∈E 0 v∈E 0 ∞ X X −βk k=0 w∈E 0 ∞ X X

w∈E 0

=

e−βk ǫs(µ)

X

w∈E 0

k=0

e−βk (B k ǫ)

w

(1 − e−β B)−1 ǫ w .

But (1 − e−β B)−1 ǫ w is the number denoted by mw in [10, Theorem 3.1], and hence the result follows from Equation (3.3) in the proof of [10, Theorem 3.1].

Proof of Proposition 7.9. Suppose that y β · ǫ = 1. Then kτ k1 = k kǫk−1 1 ǫk1 = 1, so τ is a normalised trace. Lemma 7.10 implies that −1 β −1 Z(β, τ ) = kǫk−1 1 Z(β, ǫ) = kǫk1 (y · ǫ) = kǫk1 .

Comparing (7.7) and (7.8) shows that φǫ and ψβ,τ agree on all elements sκ s∗λ , and hence by linearity and continuity agree on all of T (G, E) = T C ∗ (E). Suppose next that kτ k1 = 1 and ǫ = Z(β, τ )−1 τ . Then kǫk1 = kZ(β, τ )−1 τ k1 = Z(β, τ )−1 , so τ = kǫk−1 1 ǫ, and Lemma 7.10 gives y β · ǫ = Z(β, τ )−1 (y β · τ ) = 1.

The previous part now gives φǫ = ψβ,τ .

34


8. KMS states at the critical inverse temperature Suppose that E is a finite directed graph with no sources and that G is a groupoid which acts self-similarly on E. We consider the ideal I in T (G, E) generated by the gap projections o n X se s∗e : v ∈ E 0 , (8.1) P := pv − e∈vE 1

and the quotient O(G, E) := T (G, E)/I, which we call the Cuntz-Pimsner algebra of (G, E). The gap projections are all fixed by the action σ of R on T (G, E), and hence σ induces a dynamics on O(G, E), which we also denote by σ.

Proposition 8.1. Suppose that E is a finite graph with no sources, that (G, E) is a selfsimilar groupoid action, and that φ is a KMSβ state of (T (G, E), σ) for some β ≥ ln ρ(B). 0 Define mφ ∈ [0, 1]E by mφv := φ(pv ) for v ∈ E 0 . Then φ factors through O(G, E) if and only if Bmφ = eβ mφ . Proof. Suppose that φ factors through the quotient map q of T (G, E) onto O(G, E). Then (q(se ), q(pv )) is a Cuntz-Krieger E-family in O(G, E), and hence there is a homomorphism πq◦s,q◦p : C ∗ (E) → O(G, E). This homomorphism is equivariant for the dynamics α : R → Aut C ∗ (E) studied in [10] and the dynamics σ on O(G, E), and thus φ ◦ πq◦s,q◦p is a KMSβ state on (C ∗ (E), γ). Thus [10, Proposition 2.1] implies that Bmφ = eβ mφ . Conversely, suppose that Bmφ = eβ mφ . Then by [10, Proposition 2.1], the composition φ ◦ πq◦s,q◦p factors through a state of C ∗ (E), and hence vanishes on the gap projections in T C ∗ (E). Thus φ vanishes on the set P in (8.1). Since the elements of P are fixed by the action σ, we can apply [10, Lemma 2.2] to the set of analytic elements F = sµ ug s∗ν : µ, ν ∈ E ∗ , g ∈ G and s(µ) = g · s(ν) , and deduce that φ factors through a state of O(G, E).

The inspiration for our next result comes from [17, Proposition 7.2].

Proposition 8.2. Suppose that E is a strongly connected finite graph, and that (G, E) is a self-similar groupoid action. Then the vertex matrix B is irreducible, and has a unique 0 unimodular Perron-Frobenius eigenvector x ∈ (0, ∞)E . For g ∈ G \ E 0 , v ∈ E 0 and k ≥ 0, we define Fgk (v) := {µ ∈ d(g)E k v : g · µ = µ and g|µ = v}, and X |Fgk (v)|xv . cg,k := ρ(B)−k v∈E 0

Then for each g ∈ G \ E 0 , the sequence {cg,k : k ∈ N} is increasing and converges. The limit cg belongs to [0, xd(g) ). Proof. That E is strongly connected says precisely that B is irreducible in the sense that for all v, w ∈ E 0 , there exists n such that B n (v, w) 6= 0. Then the Perron-Frobenius theorem says that B has a unique eigenvector x with strictly positive entries such that kxk1 = 1 (see [4, Theorem 2.6] or [25, Theorem 1.6]). Now we fix g ∈ G \ E 0 and show that {cg,k } is increasing. For µ ∈ Fgk (v) and e ∈ vE 1 , we have g · (µe) = µ(g|µ · e) = µ(v · e) = µe

and g|µe = (g|µ )|e = v|e = s(e),


so µe ∈ Fgk+1 (s(e)). Thus |Fgk+1 (v)| ≥

P

cg,k+1 = ρ(B)−(k+1)

w∈E 0

X

v∈E 0

≥ ρ(B)−(k+1) = ρ(B)−(k+1)

|Fgk+1(v)|xv

v∈E 0 w∈E 0

w∈E 0

= cg,k .

|Fgk (w)|B(w, v), and we compute

X X X

35

|Fgk (w)|B(w, v)xv

|Fgk (w)|ρ(B)xw

Next we find an upper bound for the sequence {cg,k }. Since g ∈ / E 0 and G acts faithfully on TE , the action of g on d(g)E ∗ is nontrivial, and there exist j ∈ N and µ ∈ d(g)E j such that g · µ 6= µ. So µ ∈ / Fgj (s(µ)), and for every µ′ ∈ s(µ)E k−j with k ≥ j, the path µµ′ is not in Fgk (d(g)). Thus for k ≥ j and v ∈ E 0 , we have |Fgk (v)| ≤ |d(g)E k v| − |s(µ)E k−j v| = B k (d(g), v) − B k−j (s(µ), v).

The vector x is also the Perron-Frobenius eigenvector of the powers of B, and we have ρ(B k ) = ρ(B)k . Thus for k ≥ j we have X X (8.2) B k−j (s(µ), v)xv B k (d(g), v)xv − cg,k ≤ ρ(B)−k v∈E 0 k

v∈E 0

= ρ(B)−k (B x)d(g) − (B k−j x)s(µ)

= ρ(B)−k ρ(B k )xd(g) − ρ(B k−j )xs(µ) = xd(g) − ρ(B)−j xs(µ) .

The inequality (8.2) shows, first, that the increasing sequence {cg,k : k ∈ N} is bounded above, and hence converges, say to cg . Since the Perron-Frobenius eigenvector has strictly positive entries, (8.2) also shows that cg,k is bounded away from xd(g) , and hence the limit cg belongs to [0, xd(g) ). We can now state our main result about KMS states at the critical inverse temperature. Theorem 8.3. Suppose that E is a strongly connected graph with vertex matrix B, and (G, E) is a self-similar groupoid action. Let {cg : g ∈ G} be the numbers described in Proposition 8.2. (1) There is a KMSln ρ(B) state of (O(G, E), σ) such that ( ρ(B)−|κ| cg if κ = λ and d(g) = c(g) = s(κ) (8.3) ψ(sκ ug s∗λ ) = 0 otherwise. (2) Suppose that for every g ∈ G \ E 0 , the set {g|µ : µ ∈ d(g)E ∗} is finite. Then the state in part (1) is the only KMS state of (O(G, E), σ). To find KMS states at the critical inverse temperature βc , we take a sequence of KMSβ states for β > βc and take limits as β decreases to βc . We will use the Perron-Frobenius eigenvector x (which determines the numbers cg ) to get a trace τ on C ∗ (G), build a KMS state ψβ,τ using Theorem 6.1, and then take limits. However, traces on C ∗ (G) satisfy τ (id(g) ) = τ (ig−1 ig ) = τ (ig ig−1 ) = τ (ic(g) ), so to get a trace from x we need it to satisfy xd(g) = xc(g) . Fortunately, for us this is automatic because G acts self-similarly on E:

36


Proposition 8.4. Suppose that E is a strongly connected finite graph with no sources, and that (G, E) is a self-similar groupoid action. Let x be the unimodular Perron-Frobenius eigenvector of the vertex matrix B. Then xv = xw whenever v = g · w (or equivalently, whenever v = d(g) and w = c(g)). We will show that the existence of the self-similar action on E puts constraints on the vertex matrix B. We consider the set E 0 /∼ of equivalence classes for the relation on E 0 defined by v ∼ w ⇐⇒ there exists g ∈ G such that d(g) = v and c(g) = w

(see also §7). We write [v] for the class of v ∈ E 0 in E 0 /∼.

Lemma 8.5. Suppose that C, D ∈ E 0 /∼. Then for v1 , v2 ∈ C, we have X X B(v1 , w) = B(v2 , w). w∈D

w∈D

Proof. Since v1 and v2 are in the same component, there exists g ∈ G such that v1 = d(g) and v2 = c(g). Suppose that e ∈ v1 E 1 D. Then g · e ∈ v2 E 1 D, because s(g · e) = g|e · s(e) ∼ s(e) ∈ D. So g : v1 E 1 D → v2 E 1 D. We similarly have g −1 : v2 E 1 D → v1 E 1 D and g −1 · (g · e) = (g −1g) · e = e, so g is a bijection of v1 E 1 D onto v2 E 1 D. Thus X X B(v1 , w) = |v1 E 1 D| = |v2 E 1 D| = B(v2 , w). w∈D

w∈D

Proof of Proposition 8.4. We define a matrix R = (rCD ) over E 0 /∼ by rCD = |vE 1 D| for v ∈ C; Lemma 8.5 shows rCD is well-defined, independent of v ∈ C. Since paths µ ∈ E ∗ give nonzero entries in R|µ| , R is an irreducible nonnegative matrix. Let z be 0 a Perron-Frobenius eigenvector for R, and define y ∈ [0, ∞)E by yv := z[v] . Then for v ∈ C, we have X X X (By)v = B(v, w)yw = B(v, w)zD w∈D

=

X

D∈E 0 /∼

D∈E 0 /∼ w∈D

|vE 1 D|zD =

= ρ(R)zC = ρ(R)yv .

X

rCD zD

D∈E 0 /∼

In other words, y is an eigenvector of B with eigenvalue ρ(R). Since z is a PerronFrobenius eigenvector of R, we have zD > 0 for all D, and hence yv > 0 for all v. Thus y is a strictly positive eigenvector for B, and hence must be a scalar multiple tx of the unimodular Perron-Frobenius eigenvector x for B. But now if v = g · w, we have [v] = [w] and xv = tz[v] = tz[w] = xw . Now for the proof of Theorem 8.3(1), we take x to be the unimodular Perron-Frobenius eigenvector of B, and for C ∈ E 0 /∼ we write xC for the common value of {xv : v ∈ C}. For each C, we choose a representative v ∈ C, and apply Corollary 7.6 to the state τe on ′ C ∗ (Ge ) to get a trace τe,C on C ∗ (G). Then X ′ τx := xC τe,C C∈E 0 /∼


P

37

P

is a tracial state on C ∗ (G). (The normalisation works because 1 = v xv = C |C|xC .) Now for each β > ln ρ(B), Theorem 6.1 gives us a KMSβ state ψβ,τx satisfying (6.2). The formula ( xC if ig = iw for some w ∈ C ⊂ G0 (8.4) τx (ig ) = 0 if g ∈ / G0 shows that τx is independent of the choice of representatives v ∈ C. We next look at the normalising factor Z(β, τx ) in the formula (6.2) for ψβ,τx . Lemma 8.6. For β > ln ρ(B) and x, τx as above, we have −1 (8.5) Z(β, τx ) = 1 − e−β ρ(B) .

Proof. Writing the inside sum in terms of the vertex matrix gives ∞ X X Z(β, τx ) = e−βj τx (is(µ) ) =

j=0 µ∈E j ∞ X X j=0

e−βj B j (v, w)τx (iw ) .

X

v∈E 0 w∈E 0

Since Proposition 8.4 gives xw = xC for all x ∈ C, we deduce from (8.4) that ∞ X XX B j (v, w)xw e−βj Z(β, τx ) = = =

v∈E 0 j=0 ∞ XX

v∈E 0 j=0 ∞ XX

w∈E 0

e−βj (B j x)v e−βj ρ(B)j xv

v∈E 0 j=0

=

X

v∈E 0

1 − e−β ρ(B)

= 1 − e−β ρ(B)

−1

−1

xv

because eβ > ρ(B)

because kxk1 = 1.

Proof of Theorem 8.3(1). Suppose that {βn } is a decreasing sequence such that βn → β as n → ∞. If τ is any normalised trace on C ∗ (G) then running the usual weak* compactness argument on the states ψβn ,τ gives us a KMSln ρ(A) state on (T (G, E), σ) (see [16, Proposition 7.6] or [17, page 6652]). The point is that, if we take x to be the unimodular Perron-Frobenius eigenvector of B and τ = τx , then we can compute the limit of {ψβn ,τx }. We now take a spanning element sκ ug s∗λ for T (G, E), and compute lim ψβn ,τx (sκ ug s∗λ ).

n→∞

Since everything vanishes otherwise, we assume that κ = λ, s(κ) = d(g) and d(g) = c(g). Putting the formula (8.5) into (6.2) gives ∞ X X e−βn j τx ig|µ ψβn ,τx (sκ ug s∗κ ) = 1 − e−βn ρ(B) j=|κ|

{µ∈s(κ)E j−|κ| : g·µ=µ}

38

MARCELO LACA, IAIN RAEBURN, JACQUI RAMAGGE, AND MICHAEL F. WHITTAKER −βn |κ|

−βn

1−e

=e

ρ(B)

∞ X

e−βn k

k=0

X

{µ∈d(g)E k

τx ig|µ

: g·µ=µ}

.

Now we recall from (8.4) that τx (ig|µ ) vanishes unless g|µ = v for some vertex v (strictly speaking, unless g|µ is the identity morphism at v), and then τx (iv ) = xv . Thus the only µ which contribute to the sum are those which satisfy g · µ = µ and g|µ = v, which means |µ| that µ belongs to the set Fg (v) of Proposition 8.2. Thus we have ∞ X X |Fgk (v)|xv e−βn k ψβn ,τx (sκ ug s∗κ ) = e−βn |κ| 1 − e−βn ρ(B) k=0

= e−βn |κ|

∞ X

1 − e−βn ρ(B)

k=0

v∈E 0

ρ(B)−k

X

v∈E 0

|Fgk (v)|xv

k e−βn ρ(B) .

Now, modulo the factor e−βn |κ| , which converges to ρ(B)−|κ| as n → ∞, P we are in the situation of [17, Lemma 7.4], with r = e−βn ρ(B) and ck = cg,k = ρ(B)−k v∈E 0 |Fgk (v)|xv , which by Proposition 8.2 increases to cg as n → ∞. So by [17, Lemma 7.4] we have ψβn ,τx (sκ ug s∗κ ) → ρ(B)−|κ| cg as n → ∞.

Thus the limiting KMSln ρ(B) state ψ satisfies (8.3). To see that ψ factors through O(G, E), we use again the canonical homomorphism πs,p : T C ∗ (E) → T (G, E). Then ψ ◦ πs,p is a KMSln ρ(B) state of T C ∗ (E), and hence by [10, Theorem 4.3(a)] factors through C ∗ (E). By [10, Proposition 4.1], the vector mψ = mψ◦πs,p satisfies Bmψ = ρ(B)mψ = eln ρ(B) mψ , and Proposition 8.1 implies that ψ factors through O(G, E).

To start the proof of Theorem 8.3(2), we suppose that φ is a KMSln ρ(B) state of (O(G, E), σ). Then Proposition 8.1 implies that mφ = φ(pv ) satisfies Bmφ = ρ(B)mφ . P Since v pv is the identity in O(G, E), mφ is the unimodular Perron-Frobenius eigenvector of B. Thus φ(pv ) = xv = ψ(pv ) for all v ∈ E 0 . Since both φ and ψ are KMSln ρ(B) states, they are determined by their values on the generators {ug : g ∈ G} (see Proposition 5.1(2)). Thus it suffices for us to prove that φ(ug ) = cg = ψ(ug ) for all g ∈ G \ E 0 . So we fix such a g. To motivate our arguments, we start our calculation, following the P argument on [17, page 6654]. Suppose that n ∈ N. We use the Cuntz relations pv = µ∈vE n sµ s∗µ and the P decomposition 1 = v∈E 0 pv to get X X X sµ s∗µ = φ(sg·µ ug|µ s∗µ ). φ(ug ) = φ ug v∈E 0 µ∈vE n

µ∈d(g)E n

Proposition 5.1 implies that φ(sg·µug|µ s∗µ ) = 0 unless g · µ = µ. So we introduce and then (5.3) gives (8.6)

φ(ug ) =

Gng (v) := {µ ∈ d(g)E n v : g · µ = µ},

X X

ρ(B)−n φ(ug|µ )

v∈E 0 µ∈Gn g (v)

=

X

X

n v∈E 0 µ∈Gn g (v)\Fg (v)

ρ(B)−n φ(ug|µ ) +

X X

v∈E 0 µ∈Fgn (v)

ρ(B)−n φ(uv )


= ρ(B)−n

(8.7)

X

X

φ(ug|µ ) + ρ(B)−n


X

v∈E 0

39

|Fgn (v)|xv .

We are aiming to prove that φ(ug ) = cg , and the second summand in (8.7) goes to cg as n → ∞. So we need to show that the first summand goes to 0. Now we recall that since φ is a KMS state, φ(ug|µ ) = 0 unless c(g|µ) = d(g|µ) = s(µ) = v, say. In that case, ug|µ belongs to the corner D = pv πu (C ∗ (G))pv ; since φ|D is a positive functional with norm φ|D (1D ) = φ(pv ), we have |φ(ug|µ )| ≤ φ(pv ). Thus X X X Gng (v) \ Fgn (v) φ(pv ) (8.8) φ(ug|µ ) ≤ ρ(B)−n ρ(B)−n v∈E 0


= ρ(B)−n

X Gn (v) \ F n (v) xv . g g

v∈E 0

At this point we invoke the finite-state hypothesis, which implies that for fixed g ∈ G \ E 0 , the set R := g|µ : µ ∈ d(g)E ∗, g|µ 6= ids(µ) is finite. Then there exists j such that for all h ∈ R, there exists ν h ∈ d(h)E j such that h · ν h 6= ν h . Since E is strongly connected and g|µ · (νλ) = (g|µ · ν)(g|µν · λ), we can by making ν h longer (and j larger) assume that ν h ∈ d(h)E j d(h). Lemma 8.7. With the preceding notation, we have X X n Gnj (v) \ F nj (v) xv ≤ ρ(B)j − 1 xv (8.9) g g v∈E 0

v∈E 0

for every n ≥ 0.

Proof. We prove this by induction on n. For n = 0, we we have Fgnj (v) = ∅ because P g ∈ / E 0 , and both sides collapse to v xv = 1. Suppose that (8.9) holds for n, and (n+1)j (n+1)j consider λ ∈ Gg (v) \ Fg (v). We can factor λ = µµ′ with µ ∈ d(g)E nj w and µ′ ∈ wE j v for some w ∈ E 0 . Then g · λ = λ, so g · µ = µ and g|µ · µ′ = µ′ . Similarly, nj g|λ = (g|µ )|µ′ is not ids(λ) = ids(µ′ ) = v and g|µ 6= ids(µ) . Thus µ ∈ Gnj g (w) \ Fg (w) and µ′ ∈ Gjg|µ (v) \ Fg|j µ (v). Thus X X X nj nj Gg (w) \ Fgnj (w) Gj (v) \ F j (v) xv . Gnj g (w) \ Fg (w) ≤ g|µ g|µ v∈E 0 w∈E 0

w∈E 0

nj For each µ ∈ Gnj g (w) \ Fg (w), which is not in Gjg|µ , so

we have d(g|µ) = s(µ) = w, and there is a path ν ∈ wE j w

X X X j Gnj (w) \ F nj (w) G(n+1)j (v) \ F (n+1)j (v) xv ≤ B (w, v)x − x v w g g g g v∈E 0

w∈E 0

v∈E 0

X j nj Gnj = g (w) \ Fg (w) ρ(B) xw − xw w∈E 0

= ρ(B)j − 1)

X Gnj (w) \ F nj (w) xw g g

w∈E 0

Now the inductive hypothesis gives X X G(n+1)j (v) \ F (n+1)j (v) xv ≤ ρ(B)j − 1)n+1 xw , g g v∈E 0

w∈E 0

40


as required.

End of the proof of Theorem 8.3(2). We start with the formula (8.6), and estimate: |φ(ug ) − ψ(ug )| = |φ(ug ) − cg | X X = ρ(B)−n


≤ ρ(B)−n ≤

X

v∈E 0

X |Fgnj (v)|xv − cg φ(ug|µ ) + ρ(B)−n v∈E 0

X X nj Gn (v) \ F n (v) xv + ρ(B)−n |Fg (v)|xv − cg g g

v∈E 0

1 − ρ(B)−j

n

by (8.8)

v∈E 0

X |Fgnj (v)|xv − cg by Lemma 8.7. xv + ρ(B)−n v∈E 0

Now the first summand goes to 0 as n → ∞ because ρ(B) ≥ 1 (see the appendix in [10]), and the second goes to 0 by Proposition 8.2. So φ(ug ) = ψ(ug ), and φ = ψ by Proposition 5.1. This completes the proof of Theorem 8.3(2). Since we know from Proposition 5.1 that the restrictions of KMS states to the copy of C (G) in O(G, E) are traces, existence of the KMSln ρ(B) state gives the following groupoid version of [17, Corollary 7.5]. ∗

Corollary 8.8. Suppose that G is a finite groupoid that acts self-similarly on a finite graph E with no sources and vertex set G0 . Then there is a trace τ on C ∗ (G) such that τ (ug ) = cg for all g ∈ G \ G0 . 9. Computing KMS states at the critical inverse temperature Suppose that E is a strongly connected finite directed graph, that (G, E) is a self-similar groupoid action, and that for every g ∈ G \ E 0 , the set {g|µ : µ ∈ d(g)E ∗ } is finite. Then Theorem 8.3(2) implies that (O(G, E), σ) has a unique KMS state given by (8.3). We say that (G, E) contracts to a finite subset N of G if for every g ∈ G there exists n ∈ N such that {g|µ : µ ∈ d(g)E ∗, |µ| ≥ n} ⊂ N . Then the Moore diagram for N is the labelled directed graph with vertex set N and, for each g ∈ N and e ∈ d(g)E 1 , an edge from g ∈ N to g|e ∈ N labelled (e, g · e). For g ∈ N and e ∈ d(g)E 1, a typical edge in a Moore diagram looks like g

(e,g·e)

g|e

The self-similarity relations for the set N can be read off the Moore diagram: the edge above encodes the relation g · (eµ) = (g · e)(g|e · µ) for µ ∈ s(e)E ∗ . Remark 9.1. We do not assume that the contracting set N is minimal, as is often done in the literature on self-similar groups. However, as in [17, §2], we can decide whether such a set is minimal by looking for cycles in the Moore diagram of G: if g lies on a cycle, then g|µ ∈ N for all µ ∈ s(g)E ∗ . Now suppose that ψ is the KMSln ρ(B) state of (O(G, E), σ) from Theorem 8.3(1). The definition of the sets Gng (v) in (8.6) shows that X X (9.1) ρ(B)−n φ(ug|µ ) φ(ug ) = v∈E 0 µ∈Gn g (v)


41

idw (4,4) (3,1)

fb (2,3)

(2,2) (1,3)

(3,3)

(4,2)

(3,2)

idv (1,4)

fa

fb−1

(4,1)

(2,4) fa−1

(1,1)

Figure 4. The Moore diagram for the contracting set N of (9.2). =

X

ρ(B)−n φ(ug|µ ).

{µ∈d(g)E n : g·µ=µ}

Since for fixed g and large enough n, every g|µ ∈ N , the values {ψ(ug ) : g ∈ G} are determined by the values {ψ(ug ) : g ∈ N }. For g ∈ N \ E 0 , the set Fgk (v) consists of stationary paths µ ∈ d(g)E k v in the Moore diagram of the form g

(µ1 ,µ1 )

g|µ1

(µ2 ,µ2 )

g|µ1 µ2

(µ3 ,µ3 )

···

(µk ,µk )

g|µ = idv .

So we can compute the values of |Fgk (v)| by counting the stationary paths of length k from g to idv . Once we have the values |Fgk (v)|, we compute ψ(ug ) = cg by evaluating the limit limk→∞ cg,k of Proposition 8.2. We return to the graph E of Example 3.10, and the faithful self-similar groupoid action (G, E) obtained by applying Theorem 3.9 to the partial automorphisms fa and fb descibed in that example. Looking at the drawing of E in Figure 3 shows that the graph E is strongly connected. Proposition 9.2. The self-similar groupoid action (G, E) of Example 3.10 contracts to N := {idv , idw , fa , fa−1 , fb , fb−1 }.

(9.2)

Proof. All reduced elements of length two in G belong to either Rv := {fb fa , fa−1 fb−1 } or Rw := {fa fb , fa−1 fb−1 }. We compute all of their restrictions by length one paths: (9.3)

(fb fa )|1 = fa (fb fa )|2 = fb

(fa−1 fb−1 )|1 = fb−1 (fa−1 fb−1 )|2 = fa−1

(fa fb )|3 = idv (fa fb )|4 = fb fa

(fb−1 fa−1 )|3 = fa−1 fb−1 (fb−1 fa−1 )|4 = idv

The restriction of each element of Rv belongs to {fa , fb , fa−1 , fb−1 } and the restriction of each element of Rw belongs to Rv ∪ E 0 . Thus for any g ∈ Rv ∪ Rw and µ ∈ d(g)E 2 , we have g|µ ∈ N . So if g ∈ G is the product of n generators, then g|ν belongs to N for all ν ∈ d(g)E 2(n−1) . Thus (G, E) contracts to N . We draw the Moore diagram for the contracting set N of (G, E) in Figure 4.

42


e1,2,0 e1,1,1

e1,1,0

1

e2,1,0

2

e2,2,1 e2,2,0

e2,1,1

Figure 5. The graph E defined by the matrix A in (9.4). Proposition 9.3. The Cuntz-Pimsner algebra (O(G, E), σ) has a unique KMSln 2 state ψ which is given on N by ( 0 for g ∈ {fa , fb , fa−1 , fb−1 }, ψ(ug ) = 1/2 for g ∈ {idv , idw }. Proof. The spectral radius of the vertex matrix B of E is ρ(B) = 2 with normalised Perron-Frobenius eigenvector (1/2, 1/2). Theorem 8.3(2) implies that there is a unique KMSln 2 state ψ and Proposition 8.1 implies that ψ(ug ) = 1/2 for g ∈ {idv , idw }. The Moore diagram in Figure 4 shows that there are no stationary paths from elements of the set S = {fa , fb , fa−1 , fb−1 } to idv or idw . Thus Fgk (v) = Fgk (w) = ∅ for all k ∈ N and g ∈ S, and ψ(ug ) = 0 for g ∈ S.

We next examine one of the self-similar groupoid actions (GA , E) of Example 7.7, for A = {a1 , a2 } and the matrices ! ! 2 1 1 0 (9.4) A= and B = . 2 2 2 1 We draw the corresponding graph E in Figure 5, and observe that E is strongly connected. The action of the generating set {fa1 , fa2 } is given by the relations fa1 · e1,1,0 µ = e1,1,1 µ fa2 · e2,1,0 µ = e2,1,0 (fa1 · µ) fa2 · e2,2,1 ν = e2,2,0 (fa2 · ν)

fa1 · e1,1,1 µ = e1,1,0 (fa1 · µ) fa2 · e2,1,1 µ = e2,1,1 (fa1 · µ)

fa1 · e1,2,0 ν = e1,2,0 ν fa2 · e2,2,0 ν = e2,2,1 ν

Proposition 9.4. We take (A, E) as above. Then the self-similar groupoid action (GA , E) contracts to N := {id1 , id2 , fa1 , fa−1 , fa2 , fa−1 }, 1 2

(9.5)

and for each connected component Ci = {i} the reduction G{i} is isomorphic to Z. Proof. We first show that (GA , E) contracts to N by showing that for each g ∈ G and sufficiently long µ ∈ d(g)E ∗ we have g|µ ∈ N . The reduced elements of length two in G are S := {fa21 , (fa21 )−1 , fa22 , (fa22 )−1 }. We compute (9.6)

fa21 |e1,1,0 = fa1 fa22 |e2,1,0 = fa21

fa21 |e1,1,1 = fa1 fa22 |e2,1,1 = fa21

fa21 |e1,2,0 = id2 fa22 |e2,2,0 = fa2

fa22 |e2,2,1 = fa2


43

(e2,1,0 ,e2,1,0 ) (e1,1,1 ,e1,1,0 )

fa 2

fa 1

(e2,2,1 ,e2,2,0 )

(e2,1,1 ,e2,1,1 ) (e1,2,0 ,e1,2,0 ) (e1,1,0 ,e1,1,1 )

(e2,2,0 ,e2,2,1 ) (e1,2,0 ,e1,2,0 )

(e1,1,0 ,e1,1,0 )

id1

(e2,1,0 ,e2,1,0 )

(e2,2,1 ,e2,2,1 ) id2

(e2,2,0 ,e2,2,0 )

(e1,1,1 ,e1,1,1 ) (e2,1,1 ,e2,1,1 ) (e1,1,1 ,e1,1,0 )

(e2,2,1 ,e2,2,0 ) (e1,2,0 ,e1,2,0 ) (e2,1,0 ,e2,1,0 )

(e1,1,0 ,e1,1,1 )

fa−1 2

fa−1 1

(e2,2,0 ,e2,2,1 )

(e2,1,1 ,e2,1,1 )

Figure 6. The Moore diagram for N of Proposition 9.4. We see that the length of fa21 |e is less than two for all e ∈ d(fa1 ) and fa21 |e′ is either fa21 or has length one for all e′ ∈ d(fa2 ). We conclude that fa2i |µ has length less than two for all µ ∈ d(fai ) with |µ| = 2. Since (fa2i )−1 |µ = fa2i |(fa2i )−1 ·µ by Proposition 3.6(4) and partial automorphisms are length preserving, we also conclude that (fa2i )−1 |µ has length ) with |µ| = 2. So for any g ∈ S and µ ∈ d(g)E 2, we have less than two for all µ ∈ d(fa−1 i that g|µ ∈ N . Thus if g ∈ GA is the product of n generators, then g|ν belongs to N for all ν ∈ d(g)E 2(n−1) . Thus (GA , E) contracts to N , as claimed. We now show that G{i} ∼ = Z. Fix i ∈ {1, 2}, and recall from Example 7.7 that either 2 ni −1 G{i} = {i, fai , fai , · · · , fai } or G{i} = {faki : k ∈ Z}. To show the latter, we suppose k ∈ N and prove that faki 6= idi . Since fai · ei,i,0 = ei,i,1 and fai · ei,i,1 = ei,i,0 , we have that faki 6= idv for k odd. So suppose k = 2n is even. As shown in (9.6) we have fa2i |ei,i,0 = fai , and hence fa4i |ei,i,0 = fa2i |fa2i ·ei,i,0 fa2i |ei,i,0 = fa2i |ei,i,0 fai = fa2i .

Continuing in this way, we find that fa2n | = fani . Considering the action of fa2n on i ei,i,0 i sufficiently long words of the form µ = ei,i,0 ei,i,0 . . . ei,i,0, we can reduce 2n by factors of 2 · µ ei,i,0 = µ(fami · ei,i,0 ) = µ ei,i,1 . Thus | = fami with m odd. Then fa2n and arrive at fa2n i i µ faki acts nontrivially on iE ∗ for all k ∈ Z, and G{i} ∼ = Z.

We have drawn the Moore diagram for N in Figure 6. Since every element in N is in a cycle, this contracting set N is in fact minimal. Note that the central horizontal layer in Figure 6 is a copy of the graph E of Figure 5 with the arrows reversed.

44

MARCELO LACA, IAIN RAEBURN, JACQUI RAMAGGE, AND MICHAEL F. WHITTAKER (e2,1,0 ,e2,1,0 ) fa 1

fa 2 (e2,1,1 ,e2,1,1 )

(e1,2,0 ,e1,2,0 ) (e1,2,0 ,e1,2,0 ) (e1,1,0 ,e1,1,0 )

id1

(e2,1,0 ,e2,1,0 )

(e2,2,1 ,e2,2,1 ) id2

(e1,1,1 ,e1,1,1 ) (e2,1,1 ,e2,1,1 ) (e1,2,0 ,e1,2,0 )

(e2,2,0 ,e2,2,0 )

(e2,1,0 ,e2,1,0 )

fa−1 1

fa−1 2 (e2,1,1 ,e2,1,1 )

Figure 7. The cut-down Moore diagram. Proposition 9.5. Keep the above notation, and consider the self-similar groupoid action (GA , E) with contracting set N as in 9.5. Then the Cuntz-Pimsner algebra (O(GA , E), σ) has a unique KMSln(2+√2) state ψ which is given on N by √ 2−1 for g = id1 ,    2 − √2 for g = id2 , √ ψ(ug ) =  for g = fa1 and g = fa−1 , 3−2 2  1  √  −1 10 − 7 2 for g = fa2 and g = fa2 .

Proof. vertex matrix √ for E is ρ(A) = √ A computation shows that the spectral radius of the √ 2 + 2 with normalised Perron-Frobenius eigenvector x = ( 2 − 1, 2 − 2). So Theorem 8.3(2) implies that there is a unique KMSln(2+√2) state ψ. Moreover, since uv = pv by √ √ Proposition 4.4(1), Proposition 8.1 implies that ψ(u1 ) = 2 − 1 and ψ(u2 ) = 2 − 2. Since only the stationary paths in the Moore diagram are used to compute nontrivial KMS states at the critical temperature, we consider stationary paths in the Moore diagram in Figure 6, which all lie in the cut-down subdiagram shown in Figure 7. We now compute ψ(ug ) for g = fa1 . Observe that there is a length one stationary path from g to id2 . Let akij denote the entries of the matrix Ak . Then the sets |Fgk (i)| are given k−1 k−1 by |Fgk (1)| = a21 and |Fgk (2)| = a22 . To compute ψ(ug ) = cg we use Proposition 8.2 to obtain X √ √ √ k−1 k−1 |Fgk (v)|xv = (2 + 2)−k a21 cg,k = ρ(B)−k ( 2 − 1) + a22 (2 − 2) . v∈E 0

√ Then a bit of algebra gives that ψ(ug ) = cg = limk→∞ cg,k = 3√− 2 2. Since the computation for ψ(ug−1 ) is symmetric, we also have ψ(ug−1 ) = 3 − 2 2. We now compute ψ(uh ) for h = fa2 . We use (9.1) and the two stationary paths from fa2 to fa1 to compute √ √ ψ(uh ) = 2(2 + 2)−1 ψ(ug ) = 10 − 7 2. √ By symmetry, we also have ψ(uh−1 ) = 10 − 7 2.


45

Appendix A. Exel-Pardo Actions In this appendix we show that if the self-similar groups on graphs of Exel and Pardo [8] satisfy a certain faithfulness condition, then they give rise to faithful self-similar groupoid actions. We begin by recalling Exel and Pardo’s definition of a self-similar group on a graph [8, §2]. A tuple (K, E, σ, φ) is a self-similar group on a graph if K is a group and E is a graph with no sources such that (1) There is a group action σ : K → Aut E such that σk0 : E 0 → E 0 and σk1 : E 1 → E 1 are bijections, for k ∈ K, satisfying (A.1) (A.2) (A.3)

r ◦ σk1 = σk0 ◦ r,

s ◦ σk1 = σk0 ◦ s, and σk σl = σkl .

(2) There is a group 1-cocycle φ : K × E 1 → K satisfying (A.4)

φ(kl, e) = φ(k, σl1 (e))φ(l, e).

(3) For k ∈ K and e ∈ E 1 there is a standing assumption that (A.5)

0 σφ(k,e) = σk0 .

Proposition A.1 ([8, Proposition 2.4]). Suppose (K, E, σ, φ) is an Exel-Pardo self-similar group on a graph. The pair (σ, φ) define a unique pair (σ ∗ , φ∗ ) that extend (σ, φ) such that σ ∗ is an action of K on E ∗ and φ∗ is a group one-cocycle φ : K × E ∗ → K such that, for every n ≥ 0, every k ∈ K, and every v ∈ E 0 , (1) σk∗ (E n ) ⊂ E n , (2) r ◦ σk∗ = σk ◦ r on E n , (3) s ◦ σk∗ = σk ◦ s on E n , (4) σφ∗ (k,µ) (v) = σk (v) for all µ ∈ E n , (5) σk∗ (µν) = σk∗ (µ)σφ∗∗ (k,µ) (ν) where µν ∈ E n , (6) φ∗ (k, µν) = φ∗ (φ∗ (k, µ), ν) where µν ∈ E n . If E has no sources, Exel and Pardo construct a unital C ∗ -algebra OK,E as the universal algebra generated by elements {uk : k ∈ K}, mutually orthogonal projections {pv : v ∈ E 0 }, and partial isometries {se : e ∈ E 1 } subject to the conditions that • {pv : v ∈ E 0 } ∪ {se : e ∈ E 1 } is a Cuntz-Krieger E-family • the map u : K → O(K,E) : k 7→ uk is a unitary representation of K • uk se = sσk1 (e) uφ(k,e), for every k ∈ K, and e ∈ E 1 , and • uk pv = pσk0 (v) uk , for every k ∈ K, and v ∈ E 0 Proposition A.2. Suppose (K, E, σ, φ) is an Exel-Pardo self-similar group on a graph and let K×E 0 := {(k, v)} be the transformation groupoid with d(k, v) = v, c(k, v) = σk0 (v), and unit space E 0 , and consider the maps (A.6) (A.7)

(k, v) · µ := σk∗ (µ)

for µ ∈ vE ∗ and

(k, v)|µ := (φ∗ (k, µ), s(µ))

for µ ∈ vE ∗ .

Then (A.6) defines a groupoid action of K × E 0 on TE by partial isomorphisms. If the groupoid action is faithful, then (K × E 0 , E) is a self-similar groupoid action.

46


Moreover, if E has no sources then the algebra O(K × E 0 , E) from Proposition 4.7 is isomorphic to the algebra OK,E defined in [8].

Proof. We begin by showing that (k, v) : d(k, v)E ∗ → c(k, v)E ∗ defines an action by partial isomorphism, as in Definition 3.1. Proposition A.1(1) shows that if µ ∈ d(k, v)E n , then (k, v) · µ ∈ c(k, v)E n . We now show that (k, v) : d(k, v)E n → c(k, v)E n is a bijection by induction on n. The result holds for n = 1 since σk1 : E 1 → E 1 is a bijection and σk∗ extends σk1 . So assume (k, v) : d(k, v)E n → c(k, v)E n is a bijection for all (k, v) ∈ K × E 0 . For injectivity suppose that (k, v) · (eµ) = (k, v) · (f ν) for e, f ∈ d(k, v)E 1 , µ ∈ s(e)E n and ν ∈ s(f )E n . Using Definition A.1(5) we compute (k, v) · (eµ) = (k, v) · (f ν) =⇒ σk∗ (eµ) = σk∗ (f ν)

=⇒ σk∗ (e)σφ∗∗ (k,e) (µ) = σk∗ (f )σφ∗∗ (k,f ) (ν)

=⇒ σk∗ (e) = σk∗ (f ) and σφ∗∗ (k,e) (µ) = σφ∗∗ (k,f ) (ν) =⇒ (k, v) · e = (k, v) · f and (k, v)|e · µ = (k, v)|f · ν =⇒ e = f and µ = ν, where the final implication holds by the inductive hypothesis. Hence eµ = f ν, thereby proving injectivity. For surjectivity, suppose f ∈ c(k, v)E 1 and ν ∈ s(f )E n . By the induction hypothesis there exists a unique e ∈ d(k, v)E 1 satisfying (k, v) · e = f and a unique µ ∈ d((k, v)|e )E n satisfying (k, v)|e · µ = ν. To conclude the proof of surjectivity we need to show that s(e) = r(µ) so that eµ is a path in E n+1 . For this, we compute (k, v)|e · s(e) = (φ∗ (k, e), s(e)) · s(e) = σφ∗∗ (k,e) (s(e))

= =

σk∗ (s(e)) s(σk∗ (e))

by (A.7)

by (A.6)

by Proposition A.1(4) by Proposition A.1(3)

= s(f ) = r(ν) = r(σk∗ (µ)) = =

σk∗ (r(µ)) by σφ∗∗ (k,e) (r(µ)) ∗

Proposition A.1(2) by Proposition A.1(4)

= (φ (k, e), s(e)) · r(µ)

= (k, v)|e · r(µ)

by hypothesis

by (A.6)

by (A.7).

Applying (k, v)|−1 e to both sides of (k, v)|e · s(e) = (k, v)|e · r(µ) implies that s(e) = r(µ) as desired. Thus (k, v) : s(k, v)E ∗ → r(k, v)E ∗ is a bijection satisfying the first statement in (3.1). The second statement in (3.1) follows straight from Proposition A.1(5). Thus (k, v) : d(k, v)E ∗ → c(k, v)E ∗ is a partial isomorphism. It follows from equation (A.4) that ((k, v), µ) → (k, v) · µ is an action of the groupoid K × E. Now suppose the action by partial isomorphism is faithful. To show (K × E 0 , E) is a self-similar groupoid action we prove that (3.2) of Definition 3.3 holds. Suppose (k, v) ∈ K × E 0 and e ∈ d(k, v)E 1. Then for h := (k, v)|e and µ ∈ s(e)E ∗ we have (k, v) · eµ = σk∗ (eµ)

by (A.6)

= σk∗ (e)σφ∗∗ (k,e) (µ) by Proposition A.1(5) = (k, v) · e (φ∗ (k, e), s(e)) · µ by (A.6)


= (k, v) · e (k, v)|e · µ = (k, v) · e h · µ ,

47

by (A.7)

as desired. For the final assertion note that there is a direct correspondence between the projections and partial isometries in the algebras. We claim that we can embed O(KP × E 0 , E) in OK,E by defining u(k,v) := uk pv and OK,E in O(K × E 0 , E) by defining uk := v∈E 0 u(k,v) . We leave the details as an exercise for the reader. Remark A.3. Suppose that E is a directed graph and G is a groupoid with unit space E 0 acting self-similarly on TE . By Lemma 3.4(2), for g ∈ G we have

(A.8)

s(g · e) = g|e · s(e) for all e ∈ d(g)E 1.

Both Exel–Pardo [8] and B´edos–Kaliszewski–Quigg [2] consider self-similar automorphisms of TE built from graph automorphisms of E. Such automorphisms satisfy (A.9)

s(g · e) = g · s(e) for all e ∈ E 1 .

Combining the requirements (A.8) and (A.9) leads to the standing assumptions g|e · s(e) = g · s(e) for all e ∈ E 1 , imposed in [2], and g|µ · v = g · v

for all µ ∈ E ∗ and v ∈ E 1 , imposed in [8].

Remark A.4. There are self-similar groupoid actions (G, E) which do not come from selfsimilar group actions as in [8] or [2]. To see this, we note that every every action arising via Proposition A.1 has the properties that r(g · µ) = g · r(µ) and s(g · µ) = g · s(µ) (see Remark A.3). However, for the self-similar groupoid action (GA , E) of Example 3.10, we have fa (1) = 4 and s(fa (1)) = s(4) = v, but fa (s(1)) = fa (v) = w. References [1] Z. Afsar, A. an Huef and I. Raeburn, KMS states on C ∗ -algebras associated to local homeomorphisms, Internat. J. Math. 25 (2014), article no. 1450066 (28 pages). [2] E. B´edos, S. Kaliszewski and J. Quigg, On Exel-Pardo algebras, preprint, arXiv:1512.07302. [3] O. Bratteli and D.W. Robinson, Operator Algebras and Quantum Statistical Mechanics II, Second Edition, Springer-Verlag, Berlin, 1997. [4] J. Ding and A. Zhou, Nonnegative Matrices, Positive Operators, and Applications, World Scientific, Singapore, 2009. [5] M. Enomoto, M. Fujii and Y. Watatani, KMS states for gauge action on OA , Math. Japon. 29 (1984), 607–619. [6] R. Exel, Inverse semigroups and combinatorial C ∗ -algebras, Bull. Braz. Math. Soc. 39 (2008), 191– 313. [7] R. Exel and M. Laca, Partial dynamical systems and the KMS condition, Comm. Math. Phys. 232 (2003), 223–277. [8] R. Exel and E. Pardo, Self-similar graphs: a unified treatment of Katsura and Nekrashevych C ∗ algebras, preprint, ArXiv:1409.1107. [9] N.J. Fowler and I. Raeburn, The Toeplitz algebra of a Hilbert bimodule, Indiana Univ. Math. J. 48 (1999), 155–181. [10] A. an Huef, M. Laca, I. Raeburn and A. Sims, KMS states on the C ∗ -algebras of finite graphs, J. Math. Anal. Appl. 405 (2013), 388–399. [11] A. an Huef, M. Laca, I. Raeburn and A. Sims, KMS states on the C ∗ -algebras of reducible graphs, Ergod. Th. Dynam. Sys. 35 (2015), 2535–2558.

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[12] T. Kajiwara and Y. Watatani, KMS states on finite-graph C ∗ -algebras, Kyushu J. Math. 67 (2013), 83–104. [13] T. Katsura, A construction of actions on Kirchberg algebras which induce given actions on their K-groups, J. Reine Angew. Math. 617 (2008), 27–65. [14] M. Laca and S. Neshveyev, KMS states of quasi-free dynamics on Pimsner algebras, J. Funct. Anal. 211 (2004), 457–482. [15] M. Laca and I. Raeburn, Phase transition on the Toeplitz algebra of the affine semigroup over the natural numbers, Adv. Math. 225 (2010), 643–688. [16] M. Laca, I. Raeburn and J. Ramagge, Phase transition on Exel crossed products associated to dilation matrices, J. Funct. Anal. 261 (2011), 3633–3664. [17] M. Laca, I. Raeburn, J. Ramagge and M.F. Whittaker, Equilibrium states on the Cuntz-Pimsner algebras of self-similar actions, J. Funct. Anal. 266 (2014), 6619–6661. [18] P.S. Muhly, J. Renault and D.P. Williams, Equivalence and isomorphism for groupoid C ∗ -algebras, J. Operator Theory 17 (1987), 3–22. [19] V. Nekrashevych, Cuntz-Pimsner algebras of group actions, J. Operator Theory 52 (2004), 223–249. [20] V. Nekrashevych, Self-Similar Groups, Math. Surveys and Monographs, vol. 117, Amer. Math. Soc., Providence, 2005. [21] V. Nekrashevych, C ∗ -algebras and self-similar groups, J. Reine Angew. Math. 630 (2009), 59–123. [22] G.K. Pedersen, C ∗ -Algebras and their Automorphism Groups, London Math. Soc. Monographs, vol. 14, Academic Press, London, 1979. [23] I. Raeburn, Graph Algebras, CBMS Regional Conference Series in Mathematics, vol. 103, Amer. Math. Soc., Providence, 2005. [24] I. Raeburn and D.P. Williams, Morita Equivalence and Continuous-Trace C ∗ -Algebras, Math. Surveys and Monographs, vol. 60, Amer. Math. Soc., Providence, 1998. [25] E. Seneta, Nonnegative Matrices and Markov Chains, Second Edition, Springer-Verlag, Berlin, 1973. Marcelo Laca, Department of Mathematics and Statistics, University of Victoria, Victoria, BC V8W 3P4, Canada E-mail address: [email protected] Iain Raeburn, Department of Mathematics and Statistics, University of Otago, PO Box 56, Dunedin 9054, New Zealand E-mail address: [email protected] Jacqui Ramagge, School of Mathematics and Statistics, University of Sydney, NSW 2006, Australia. E-mail address: [email protected] Michael F. Whittaker, School of Mathematics and Statistics, University of Glasgow, University Gardens, Glasgow Q12 8QW, Scotland E-mail address: [email protected]